Answer:111
Step-by-step explanation: how to do it:is try
Answer: f(k) = 3k - 5
Step-by-step explanation:
For f(x) functions, you're simply plugging in whatever value is in the parenthesis. For example, if you were asked to find f(1), you would plug in 1 for x.
f(1) = 3(1) - 5 = -2. In this case, you're just plugging in a different variable.
Annette is standing with a tree directly between her and a building. She stands so that, in her eyes, the
top of the tree appears to "touch" the top of the building. If the distance between her and the tree is
30 ft, the distance between the tree and the building is 15 ft, and the tree is 20 ft tall, how tall is the
building? (You may ignore her height.)
The height of the building is 30 feet.
How to find the height of the building?Annette is standing with a tree directly between her and a building. She stands so that, in her eyes, the top of the tree appears to "touch" the top of the building.
The distance between her and the tree is 30 ft, the distance between the tree and the building is 15 ft, and the tree is 20 ft tall.
Therefore, the height of the building can be calculated as follows:
The situation forms two right angle triangle.
Therefore, using similar triangle ratio,
20 / 30 = x / 45
where
x = height of the buildingHence,
45 × 20 = 30x
900 = 30x
x = 900 / 30
x = 30
Therefore,
height of the building = 30 ft
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Suppose you roll the number cube 199 more times would you expect the experimental probability of rolling a 3 or 4 the same as your answer in exercise 5
I need help with elimination, the first part is 4x+5y=48 and the second part is 7x–3y=−10
The solution to the system of equations in this problem is given as follows:
(2,8).
How to solve the system of equations?The system of equations for this problem is defined as follows:
4x + 5y = 48.7x - 3y = -10.To eliminate the variable y, we multiply the first equation by 3 and the second by 5, hence:
12x + 15y = 144.35x - 15y = -50.Adding the two equations, the solution for x is obtained as follows:
47x = 94
x = 94/47
x = 2.
Hence the solution for y is obtained as follows:
4(2) + 5y = 48
5y = 40
y = 40/5
y = 8.
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Alexia used three reams of paper in her first semester of school. Each ream has the same number of sheets. She used another 40 sheets after that. Write an expression to represents the number of sheets Alexia used
An expression to represent the number of sheets Alexia used is y = 3x + 40.
In math, expressions are mathematical statements with a minimum of two terms containing numbers or variables, or both, connected by an operator in between.
The mathematical operators can be any of addition, subtraction, multiplication, or division. For example, p + q is an expression where p and q are terms having an addition operator in between.
There are two types of expressions, numerical expressions - which contain only numbers, and algebraic expressions- which contain both numbers and variables.
let
x = the number of sheets per ream
y = the total number of sheets
Alexia used three reams of paper, so the total number of sheets per ream will be 3x since each ream has the same number of sheets.
After that, she used another 40 sheets. Therefore the expression will become as
y = 3x + 40 represents the total number of sheets Alexia used.
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Find the surface area of this rectangular prism. Be sure to include the correct unit in your answer.
PLSSSSS HELPPP!!!!!
The surface area of the rectangular prism is 112 square metes.
What is surface area?The sum of all area of each surface that make up an object is referred to as its surface area.
So that in the given question,
Area of a rectangle = length x width
i. area of its base = length x width
= 8 x 2
= 16 sq. m
ii. area of its front surface = length x width
= 4 x 8
= 32 sq. m
iii. area of it one of its sides = length x width
= 4 x 2
= 8 sq. m
Therefore,
surface area of the rectangular prism = (2 x 16) + (2 x 32) + (2 x 8)
= 32 + 64 + 16
= 112
The surface area of the rectangular prism is 112 square meters.
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A three-column table is given. Part A C D Part 14 28 63 Whole 50 B 90 What is the value of B in the table? 64 50 40 12
Using the constant of proportionality the value for B in the table is obtained as option C: B = 40.
What is a COP?
The ratio between two quantities that are directly proportional is the constant of proportionality. When two quantities grow and shrink at the same rate, they are directly proportional.
The table contains three columns A, C, D.
The first row contains all the part values - 14, 28, 63
The second row contains all the whole values - 20, B, 90
It is necessary to calculate the constant of proportionality.
In this case, for A, we have 14 part and 20 whole, the constant will be -
COP = Whole / Part
COP = 20 / 14
COP = 1.42857
In this case, for C, we have 63 part and 90 whole, the constant will be -
COP = Whole / Part
COP = 90 / 63
COP = 1.42857
Since, the constant of proportionality is same for A and C, it will be same for B also.
In this case, for B, we have 28 part and B whole, the constant will be -
COP = Whole / Part
1.42857 = B / 28
B = 1.42857 × 28
B = 40
Therefore, the value for B is obtained as 40.
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A three-column table is given. Part A C D Part 14 28 63 Whole 20 B 90 What is the value of B in the table? 64 50 40 12
30 5.109 ÷ 100 in decimal
The result of the division of the number by 100 gives 3.05109 in decimal.
Define the term decimal number?A decimal is really a number that is divided into two parts: another whole and a fraction.
Between integers, decimal numbers are used to express the numerical value of complete and partially whole quantities.Then, there are two parts to a decimal number: a whole number portion and a fractional portion. The whole component of something like a decimal number has the same decimal point value system as the complete number. After the decimal point, however, when we proceed to the right, we obtain the fractional portion of the decimal number.For the given number.
= 305.109 ÷ 100
Then point will shift to 2 decimals towards left.
= 3.05109
Thus, the result of the division of the number by 100 gives 3.05109 in decimal.
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rationalize the denominator of $\displaystyle \frac{1}{\sqrt{2} \sqrt{3} \sqrt{7}}$, and write your answer in the form\[ \frac{a\sqrt{2} b\sqrt{3} c\sqrt{7} d\sqrt{e}}{f}, \]where everything is in simplest radical form and the fraction is in lowest terms, and $f$ is positive. what is $a b c d e f$?
The sum of a+b+c+d+e+f=57.
Apparently, you want to simplify: [tex]\frac{1}{\sqrt{2} +\sqrt{3} +\sqrt{7} }[/tex]
so the denominator is rational. It looks like the form you want is:
[tex]\frac{A\sqrt{2}+B\sqrt{3} +C\sqrt{7} +D\sqrt{E} }{F}[/tex]
And you want to know the sum A+B+C+D+E+F.
We can start by multiplying the numerator and denominator by a conjugate of the denominator. Then we can multiply the numerator and denominator by a conjugate of the resulting denominator.
[tex]\frac{1}{\sqrt{2} +\sqrt{3} +\sqrt{7}} .\frac{\sqrt{2} +\sqrt{3} -\sqrt{7}}{\sqrt{2} +\sqrt{3} -\sqrt{7}} =\frac{\sqrt{2} +\sqrt{3} -\sqrt{7}}{2\sqrt{6-2} } \\\\\\=\frac{\sqrt{2} +\sqrt{3} -\sqrt{7}}{2\sqrt{6} -2} .\frac{\sqrt{6}+1 }{\sqrt{6} +1} =\frac{(1+\sqrt{6} )(\sqrt{2} +\sqrt{3} -\sqrt{7})}{10} \\\\\\=\frac{\sqrt{2} +\sqrt{3} -\sqrt{7}+2\sqrt{3}+3\sqrt{2} -\sqrt{42} }{10} =\frac{4\sqrt{2} +3\sqrt{3} -\sqrt{7} -\sqrt{42} }{10}[/tex]
Comparing this to the desired form we have:
A = 4, B = 3, C = -1, D = -1, E = 42, F = 10
Then the sum is : A +B +C +D +E +F = 4 + 3 -1 -1 +42 +10 = 59 -2 = 57
The sum of interest is 57.
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Simplify
The following expression.
6 1/1 ÷ 6 91/10
A. 18
B. 1/126
C. 216
D. 1/18
A mathematical expression is a phrase that has a minimum of two numbers or variables and at least one mathematical operation.
Explain about the Expression?A term may be a number, a variable, the product of two or more variables, a number and a variable, or any combination of these. A single term or a collection of phrases can be used to create an algebraic expression. For instance, 4x and y are the two terms in the formula 4x + y.
Expressions that use numbers to perform operations. An example of a numerical expression is 2(3 + 8). At least one variable and one operation must be present in an algebraic expression (addition, subtraction, multiplication, division). One such algebraic expression is 2(x + 8y).
The following elements can be used to determine the algebraic expressions: a single variable or a group of related variables.
= 6/6 /546/60
=1 / 9.1
=0.109
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where do i put the line pls help i’ll give brainly
Answer:
5,7
Step-by-step explanation:
a tin box of 1 m length, 85cm width and 60cm depth is to be made. if the box is without top, find the area of the tin required to make the box and the cost of tin at the rate of 0.50 Rupees per sq. cm
The area of the tin box without the top is 39100 square cm.
The cost of making this tin box is Rs. 19550.
What is a cuboid?A cuboid is a three-dimensional closed figure which has volume along with surface area.
The volume of a cuboid is the product of its length, width, and height.
The total surface area of a cuboid is 2(lw + wh + wl) and the lateral surface area is 2(l + w)×h.
Given, A tin box of 1 m in length, 85cm in width, and 60cm in depth is to be made.
We know the total surface area of a cuboid is 2(lw + wh + wl).
Therefore, The surface area of the cuboid without the top would be,
= 2(lw + wh + wl) - (l×w).
= 2(100×85 + 85×60 + 85×100) - (85×60).
= 2(22100) - 5100.
= 44200 - 5100.
= 39100 square cm.
The cost of making this tin box is Rs.(39100×0.50).
= Rs. 19550.
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if an integer is randomly selected from all positive 2-digit integers, what is the probability that the integer chosen has (a) a 4 in the tens place? (b) at least one 4 in the tens place or the units place? (c) no 4 in either place?
The probability of 4 in the tens place is [tex]\frac{1}{9}[/tex], the probability of at least one 4 in the tens place or the units place is [tex]\frac{1}{5}[/tex], the probability no 4 in either place is [tex]\frac{4}{5}[/tex].
(a) total number of 2-digit terms =(99 - 10) + 1 = 90
n(s) = 90
A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%
number of 2 digits that have 4 in tens place = 40, 41, 42 ......49
n(a) = 10
P(A) =[tex]\frac{ n(a)}{n(s)}[/tex]
Substituting the values
P(A) = [tex]\frac{10}{90}[/tex]
P(A) = [tex]\frac{1}{9}[/tex]
Therefore, P(A) = [tex]\frac{1}{9}[/tex]
(b) From the previous one, we know the P(A) = [tex]\frac{1}{9}[/tex]
Number of 4's in units place = 14, 24, 34.....94 = 9
n(b) = 9
P(B) = [tex]\frac{ n(b)}{n(s)}[/tex]
Substituting the values
P(B) = [tex]\frac{9}{90}[/tex]
P(B) =[tex]\frac{1}{10}[/tex]
P(A and B) = [tex]\frac{1}{10}[/tex]
Using OR probability, we have
P(A or B) = P(A) + P(B) - P(A and B)
P([tex]\frac{1}{9}[/tex] or [tex]\frac{1}{10}[/tex]) = [tex]\frac{1}{9}[/tex] + [tex]\frac{1}{10}[/tex] - [tex]\frac{1}{90}[/tex]
P([tex]\frac{1}{9}[/tex]or [tex]\frac{1}{10}[/tex]) = [tex]\frac{1}{5}[/tex]
Therefore, P([tex]\frac{1}{9}[/tex]or [tex]\frac{1}{10}[/tex]) = [tex]\frac{1}{5}[/tex]
(c) Probability of no 4 in either place = 1 - ( probability of 4 in either place)
= 1 - [tex]\frac{1}{5}[/tex] = [tex]\frac{4}{5}[/tex]
Therefore, probability of no 4 in either place = [tex]\frac{4}{5}[/tex]
Therefore, the probability that the integer chosen has a 4 in the tens place is [tex]\frac{1}{9}[/tex], P([tex]\frac{1}{9}[/tex] or [tex]\frac{1}{10}[/tex]) = [tex]\frac{1}{5}[/tex], and probability of no 4 in either place = [tex]\frac{4}{5}[/tex].
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If the domain of f(x) is -3 ≤ x ≤ 9 and the range of f(x) is 2 ≤ y ≤ 15, then which of the following
statements is correct about the domain and range of 9 (x) = f(x-2) - 8?
"The domain of 9(x) = f(x-2) - 8 is -3 ≤ x ≤ 9 and the range is -6 ≤ y ≤ 7" is correct.
What is the Domain and Range of the function?
The domain of a function is the set of all input values (x-values) for which the function is defined. It is the set of all values of x for which the function is defined.
The range of a function is the set of all output values (y-values) that the function can produce. It is the set of all possible values that the function can take on.
The domain of a function is the set of all input values (x-values) for which the function is defined. The range of a function is the set of all output values (y-values) that the function can produce.
When we shift a function horizontally, the domain remains unchanged, but the range will change. In the case of a shift of -2, the domain of f(x) is still -3≤x≤9
For 9(x) = f(x-2) - 8, it means that we are taking the output of f(x-2) and subtracting 8 from it. So, the range of 9(x) will be (2-8) ≤ y ≤ (15-8) = -6 ≤ y ≤ 7.
In summary,
The domain of 9(x) = f(x-2) - 8 is -3 ≤ x ≤ 9
The range of 9(x) = f(x-2) - 8 is -6 ≤ y ≤ 7
Hence, statement "The domain of 9(x) = f(x-2) - 8 is -3 ≤ x ≤ 9 and the range is -6 ≤ y ≤ 7" is correct.
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what is the answer to this
The matching of the inequality with the statement will be 1-B, 2-D, 3-A, and 4-C.
What is inequality?Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.
The matching of inequality with the statement is given as,
The inequality t < 28 represents that Tia ran the race in under 28 seconds.The inequality t > 28 represents that the temperature is warmer than 28°F.The inequality t < 29 represents that Tony is younger than 29 years old.The inequality t > 29 represents that the table is heavier than 29 kilograms.The matching of the inequality with the statement will be 1-B, 2-D, 3-A, and 4-C.
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Multiply.
(x² - 5x) (2x²+x-3)
O A. 2x+9x3 - 8x² + 15x
OB. 2x4-9x3-8x² + 15x
OC. 4x+9x3-8x² + 15x
OD. 2-9x3-9x² - 15x
The resultant of the given expression (x² - 5x) (2x²+x-3) is 2x^4 - 7x^3 -8 x^2 + 15x .
What is Algebraic expression ?
Algebraic expression can be defined as the combination of variables and constants.
Given expression,
(x² - 5x) (2x²+x-3)
= x^2 ( 2x^2+x-3) - 5x(2x^2 + x - 3)
= 2x^4 + x^3 - 3x^2 - 5x * 2x^2 + x*-5x - 3 * -5x
= 2x^4 + x^3 -3x^2 - 10 x^3 - 5x^2 + 15x
= 2x^4 + x^3 - 10x^3 -3x^2 - 5x^2 + 15x
= 2x^4 - 7x^3 -8 x^2 + 15x .
Hence, The resultant of the given expression (x² - 5x) (2x²+x-3) is 2x^4 - 7x^3 -8 x^2 + 15x .
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All the information is in the picture help please
That X, w, and Z w are related we demonstrate that these are identical.
What is congruence of angle?This makes a right triangle with segment y V as the opposite side, and we once more know that X Y and segment y Z are perpendicular. The reason why these Arkan wrote them this time is because, in my mind, they are nooses of similar triangles on the triangles, X, y v um, and our other triangle will be a Z y V. Finally, we wish to demonstrate the congruence of angle visi w with angle v X w, which will be right here. So, it's a little difficult to establish this one in this instance. However, in essence, because we have previously established that X, w, and Z w are related we demonstrate that these are identical. Additionally, we demonstrated that the Zeevi Arkin Group and XVI, which border on the last side, will both inform us that our angles are same. Therefore, given that we have previously established certain elements,
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Using the vertex (5, 17) and a point (11, -109), write your equation in Vertex Form.
Answer:
y=-3.5(x−5)^2+17
Step-by-step explanation:
Let's assume that since this equation has a vertex that it is a parabola with the form y = ax^2 + bx + c.
Let's rewrite it in vertex form as:
y=a(x−h)^2+k
where h and k are the horizontal (h) and vertical (k) coordinates of the vertex.
The vertex of (5,17) gives us both h (5) and k (17), Substituting these values gives us:
y=a(x−5)^2+17
To find a, use the one given point the parabola intersects (11,-109) and solve for a:
y=a(x−5)^2+17
-109=a((11)−5)^2+17
-109=a(6)^2+17
-109-17=a(36)
36a = -126
a = -3.5
This leads us to:
y=-3.5(x−5)^2+17
See the attached graph.
Pls help me find the answer for this
The 5th graders are going to paint miniature self portraits in art class. Each student has a canvas that is 5. 3 cm wide and 8. 5 cm tall.
The art teacher has 6 bottles of white paint for the background. Each bottle covers 220. 8 square cm of canvas.
There are 30 students in the class.
Does he have enough paint for all the canvases? If not, how much more does he need?
There is not enough paint for all the canvases. 26.7 cm² of more white paint is required.
Area of each canvas = Length × width
Area of each canvas = 5.3 × 8.5
Performing multiplication
Area of each canvas = 45.05 cm²
Total area of canvas in the class = 45.05 × 30
Performing multiplication
Total area of canvas in the class = 1351.5 cm²
Available amount of white paint = 6 × 220.8
Performing multiplication
Available amount of white paint = 1324.8 cm²
The available amount is less than canvas area in the class.
Amount of paint required = 1351.5 - 1324.8
Performing subtraction
Amount of paint required = 26.7 cm²
Thus, 26.7 cm² of white paint is required.
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Which of the following is the product of the rational expressions shown below?
[tex]\frac{3x^{2} }{x^{2} +x-6}[/tex] is the product of the rational expressions.
What are rational expressions?
Two polynomials' ratio is shown through rational expressions. It denotes a polynomial in the denominator as well as the numerator. It is an algebraic expression that contains an unknown variable and is similar to a fraction in that it is a ratio.
We are given [tex]\frac{3x^{2} }{x^{2} +x-6}[/tex]
This means that the expression has to be multiplied by the number 1 for obtaining the product of the rational expression.
We know that when anything is multiplied by the number 1, we get the number itself.
Hence, [tex]\frac{3x^{2} }{x^{2} +x-6}[/tex] is the product of the rational expressions.
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find an appropriate scale to use to graph the algebraic tule y= 3x — 11. Consider input values from -5 to 5.
To graph the algebraic rule y = 3x - 11, we need to consider input values from -5 to 5 and also choose appropriate scaling for the x and y axes.
How to use scale?A common way to choose a scale is to use 1 or 2 increments on the x and y axes. This makes it easy to draw points and read charts easily.
The x-axis goes from -5 to 5, so you can scale from -5 -4 -3 -2 -1 0 1 2 3 4 5.
For the y-axis, I need to find the minimum and maximum values of y given the input x values.
Substituting the x value into the formula y = 3x - 11 gives the y value.
y = 3(-5) - 11 = -16
y = 3(5) - 11 = 14
So for the y axis you can take -20 -15 -10 -5 0 5 10 15 20
So a good scale for the x-axis is -5 to 5 in increments of 1, and -20 to 20 in increments of 5 for the y-axis. Graphing the equation y = 3x - 11 using these scales makes the graph easier to read and understand.
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Which function has a range limited to only negative numbers?
Answer:
F(x)=-x^2
Step-by-step explanation:
The one example of function with range of only negative number is:
F(x)=-x^2
Why this? We know for every x, x^2 is positive. So -x^2 will be negative.
suppose you went on a vacation in a different country and returning to your school. is the variable 'time to destination' quantitative or qualitative?
The variable 'time to destination' quantitative.
Quantitative variables are those whose values are obtained through measurement or counting. A qualitative variable, also referred to as a category variable, is a non-numerical variable. It describes information that can be categorized. Examples like Eye colors, States, Dog breeds etc.
Any variable that isn't numerical is referred to as a qualitative variable or a category variable. It describes information that falls into a category. Generally speaking, a variable is considered to be quantitative if it can be used mathematically (such as addition). Otherwise, it's top-notch. You cannot, for instance, add blue and green.
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use zero product property to determine x ints of y= x^2 +4x-20 exact and approximate
By cancelative property, the roots of quadratic equation x² + 4 · x - 20 are: Exact: - 2 + 2√6, - 2 - 2√6, Approximate: 2.899, -6.899.
How to find the roots of a polynomial by cancelative property
In this problem we must factorize a quadratic equation and use an algebraic property known as cancelative property, defined as:
If a · b = 0, then a = 0 or b = 0.
In addition, we can apply the following factorization rule:
x² - (r₁ + r₂) · x + r₁ · r₂ = (x - r₁) · (x - r₂)
First, factorize the quadratic equation:
x² - (- 2 + 2√6 - 2 - 2√6) · x + (- 2 + 2√6) · (- 2 - 2√6)
(x + 2 - 2√6) · (x + 2 + 2√6)
Second, apply cancelative property:
x + 2 - 2√6 = 0
x = - 2 + 2√6
x + 2 + 2√6 = 0
x = - 2 - 2√6
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A plane is on its approach to land on the runway. The jet’s height above the ground is given in feet as a function of the time in seconds. The following table tracks the plane as it lands:
t (in seconds)
h (in feet)
0
4000
5
3500
10
3000
15
2500
20
2000
25
1500
Is this function linear? If it is, what is the slope? Use the formula m = StartFraction delta h Over delta t EndFraction.
a.
No, this is not a linear function.
b.
Yes, this function is linear; the slope is 500 feet/second.
c.
Yes, this function is linear; the slope is 100 feet/second.
d.
Yes, this function is linear; the slope is -100 feet/second.
Answer:
d. Yes, this function is linear; the slope is -100 feet/second.
Step-by-step explanation:
Find the difference in height at two different time periods.
If
Δh = difference in height values
Δt = difference in seconds
then
[tex]\mathrm{slope = \dfrac{\Delta h}{\Delta y}}[/tex]
Take the difference in heights at t = 0 and t = 5
The height drops from 4000 to 3500 feet so Δh = 3500 - 400 = -500
Δt = 5- 0 = 5 seconds
slope = -500/5 = -100
If you compute the slope between any pair of heights and time difference you will get the same value
So slope is constant indicating a linear function
11. A C Best Buy is selling a 62" TV for $1,250. The TV is on sale for 15% off and there is a 6% sales tax. What is the final cost of the TV? $998.75 $1126.25 B D $198.75 $1523.75
Answer:
$1,684.25
Step-by-step explanation:
(.85 x 1250) = 1062.50 If we take off 15% we leave on 85% which as a decimal is .85
Now add the tax
1062.50 + .06(1062.50)
1620.50 + 63.75
1684.25
You can graph the solutions of an inequality on a number line. How can you show the solutions of the inequality 3x +2 ≥ 1/2 ( x + 5 )
A graph of the solutions of this inequality 3x + 2 ≥ 1/2(x + 5) is shown on the number line attached below.
How to graph the solutions of the inequality?In this exercise, we would solve the given inequality by making x the subject of formula as follows;
3x + 2 ≥ 1/2(x + 5)
By multiplying both sides of the inequality by 2, we have the following:
(3x + 2) × 2 ≥ 1/2(x + 5) × 2
6x + 4 ≥ x + 5
By rearranging and collecting like terms, we have the following:
6x - x ≥ 5 - 4
5x ≥ 1
x ≥ 1/5
Next, we would use an online graphing calculator to plot the inequality as shown in the graph attached below. Additionally, the circle on the number line (graph) is closed because of the greater than or equal to (≤) symbol.
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Given the function h(x)=3x, which statement is true about h(x)?
O The function is decreasing on the interval (-∞, 0).
O The function is increasing on the interval (-∞, 0).
O The function is decreasing on the interval (0, ∞).
O The function is increasing on the interval (0, ∞).
Answer:
Two statement are true.
O The function is increasing on the interval (-∞, 0).
O The function is increasing on the interval (0, ∞).
Step-by-step explanation:
The function h(x) = 3x is a linear function (straight line) with a slope of 3, and a y-intercept of 0. The constant, positive slope means that h(x) will increase over the interval (-∞, ∞).
See the attached graph.
This stuff is hard somebody help me pls
The roots of quadratic equations:
b = ± 4 a = ± 1 / 2 x = 11 / 4 ± 17 / 4 n = ± 4 x = 3 / 4 ± √(1517 / 80) x = 7 / 8 ± √(5937 / 256) k = - 7 / 10 ± √(2809 / 100) m = 3 / 2 ± √(361 / 4) n = 3 / 2 ± 4 n = 1 / 6 ± 841 / 36 p = ± 4 v = 5 / 6 ± 14 r = - 5 / 3 ± √(718 / 27) x = 11 / 10 ± √(1 / 10) n = 11 / 4 ± 13 / 4 x = ± 5How to solve quadratic numbers
In this question we need to determine sixteen cases of quadratic equations, whose roots should be determined. This can be done by following procedure:
Write the quadratic equation.Complete the square.Simplify the resulting polynomial into a perfect square trinomial.Clear the independent variable.Now we proceed to show the solution to each case:
Case 1
b² - 4 = 0
b² = 4
b = ± 4
Case 2
4 · a² - 1 = 0
4 · a² = 1
a² = 1 / 4
a = ± 1 / 2
Case 3
2 · x² = 21 + 11 · x
2 · x² - 11 · x - 21 = 0
2 · [x² - (11 / 2) · x - 21 / 2] = 0
2 · [x² - (11 / 2) · x + 121 / 16] = 2 · (121 / 16) + 2 · (21 / 2)
2 · (x - 11 / 4)² = 289 / 8
(x - 11 / 4)² = 289 / 16
x - 11 / 4 = ± 17 / 4
x = 11 / 4 ± 17 / 4
Case 4
2 · n² = 32
n² = 16
n = ± 4
Case 5
5 · x² = 3 · x + 92
5 · x² - 3 · x - 92 = 0
5 · [x² - (3 / 2) · x - 92 / 5] = 0
5 · [x² - (3 / 2) · x] = 5 · (92 / 5)
5 · [x² - (3 / 2) · x + 9 / 16] = 5 · (92 / 5 + 9 / 16)
5 · (x - 3 / 4)² = 1517 / 16
(x - 3 / 4)² = 1517 / 80
x - 3 / 4 = ± √(1517 / 80)
x = 3 / 4 ± √(1517 / 80)
Case 6
4 · x² = 7 · x + 92
4 · x² - 7 · x - 92 = 0
4 · [x² - (7 / 4) · x - 23] = 0
4 · [x² - (7 / 4) · x + 49 / 64] = 92 + 49 / 64
4 · (x - 7 / 8)² = 5937 / 64
x - 7 / 8 = ± √(5937 / 256)
x = 7 / 8 ± √(5937 / 256)
Case 7
5 · k² = - 7 · k + 138
5 · k² + 7 · k - 138 = 0
5 · [k² + (7 / 5) · k - 138 / 5] = 0
5 · [k² + (7 / 5) · k] = 5 · (138 / 5)
5 · [k² + (7 / 5) · k + 49 / 100] = 5 · (138 / 5 + 49 / 100)
5 · (k + 7 / 10)² = 2809 / 20
k + 7 / 10 = ± √(2809 / 100)
k = - 7 / 10 ± √(2809 / 100)
Case 8
m² - 88 = 3 · m
m² - 3 · m - 88 = 0
m² - 3 · m - 352 / 4 = 0
m² - 3 · m = 352 / 4
m² - 3 · m + 9 / 4 = 361 / 4
(m - 3 / 2)² = 361 / 4
m - 3 / 2 = ± √(361 / 4)
m = 3 / 2 ± √(361 / 4)
Case 9
4 · n² - 7 = 12 · n
4 · n² - 12 · n = 7
4 · (n² - 3 · n + 9 / 4) = 7 + 9
4 · (n - 3 / 2)² = 16
n - 3 / 2 = ± 4
n = 3 / 2 ± 4
Case 10
6 · n² - 2 · n = 140
6 · [n² - (1 / 3) · n] = 140
6 · [n² - (1 / 3) · n + 1 / 36] = 140 + 1 / 6
6 · (n - 1 / 6)² = 841 / 6
n - 1 / 6 = ± 841 / 36
n = 1 / 6 ± 841 / 36
Case 11
4 · p² = 64
p² = 16
p = ± 4
Case 12
6 · v² - 10 · v = 84
6 · [v² - (5 / 3) · v] = 84
6 · [v² - (5 / 3) · v + 25 / 36] = 84
6 · (v - 5 / 6)² = 84
(v - 5 / 6)² = 14
v - 5 / 6 = ± 14
v = 5 / 6 ± 14
Case 13
3 · r² - 77 = - 10 · r
3 · r² + 10 · r = 77
3 · [r² + (10 / 3) · r] = 77
3 · [r² + (10 / 3) · r + 25 / 9] = 77 + 25 / 9
3 · (r + 5 / 3)² = 718 / 9
(r + 5 / 3)² = 718 / 27
r + 5 / 3 = ± √(718 / 27)
r = - 5 / 3 ± √(718 / 27)
Case 14
5 · x² - 11 · x = - 6
5 · [x² - (11 / 5) · x + 121 / 100] = - 6 + 5 · (121 / 100)
5 · (x - 11 / 10)² = 1 / 20
(x - 11 / 10)² = 1 / 100
x - 11 / 10 = ± √(1 / 10)
x = 11 / 10 ± √(1 / 10)
Case 15
2 · n² = 11 · n + 6
2 · n² - 11 · n = 6
2 · [n² - (11 / 2) · n] = 6
2 · [n² - (11 / 2) · n + 121 / 16] = 6 + 2 · (121 / 16)
2 · (n - 11 / 4)² = 169 / 8
(n - 11 / 4)² = 169 / 16
n - 11 / 4 = ± 13 / 4
n = 11 / 4 ± 13 / 4
Case 16
4 · x² = 100
x² = 25
x = ± 5
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1. A TV has a width of 45 inches and a height of 30
inches. What is the length of the diagonal of the TV?
A TV has a width of 45 inches and a height of 30inches the length of the diagonal of the TV is 54.08inches.
What is Rectangle?
A rectangle is a quadrilateral with four right angles in the Euclidean plane. It can alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal. A square is a rectangle with four equally long sides.
Make a diagonal and a rectangle. Two parallel triangles appear. Pay attention to one of the triangles.
We have a right triangle with a = 45 and b = 30
we have to find the c = ?
Let's find b using the Pythagorean Theorem;
a^2 + b^2 = c^2
45^2 + 30^2 = c^2
2025 + 900 = c^2
2925 = c^2
c = 54.08 inches
Therefore,the diagonal of the TV is 54.08inches.
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