Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box. Also, specify any restrictions on the variable.y²-3y - 18/y²-9y + 18Rational expression in lowest terms:Variable restrictions for the original expression: y
Answers
Given: The expression below
[tex]\frac{y^2-3y-18}{y^2-9y+18}[/tex]To Determine: The lowest term of the given rational fraction
Solution
Let simplify both the numerator and the denominator
[tex]\begin{gathered} Numerator:y^2-3y-18 \\ y^2-3y-18=y^2-6y+3y-18 \\ y^2-3y-18=y(y-6)+3(y-6) \\ y^2-3y-18=(y-6)(y+3) \end{gathered}[/tex][tex]\begin{gathered} Denominator:y^2-9y+18 \\ y^2-9y+18=y^2-3y-6y+18 \\ y^2-9y+18=y(y-3)-6(y-3) \\ y^2-9y+18=(y-3)(y-6) \end{gathered}[/tex]Therefore
[tex]\begin{gathered} \frac{y^2-3y-18}{y^2-9y+18}=\frac{(y-6)(y+3)}{(y-3)(y-6)} \\ y-6-is\text{ common} \\ \frac{y^{2}-3y-18}{y^{2}-9y+18}=\frac{(y-6)(y+3)}{(y-3)(y-6)} \\ \frac{y^{2}-3y-18}{y^{2}-9y+18}=\frac{y+3}{y-3} \end{gathered}[/tex]Hence, the rational expression in its lowest term is
[tex]\frac{y+3}{y-3}[/tex]The variable for the original expression is as given as
[tex]\begin{gathered} \frac{y^{2}-3y-18}{y^{2}-9y+18}=\frac{(y-6)(y+3)}{(y-3)(y-6)} \\ y\ne3,y\ne6 \end{gathered}[/tex]Related Questions
can someone please help!!!
Answers
The simplified expression is as follows:
[tex]\frac{\frac{3x^{6} }{14y^{9} } }{\frac{33x^{4} }{10y^{2} } } = \frac{5x^{2} }{77y^{7} }[/tex]
How to simplify expression?The expression can be simplified as follows:
To simplify an expression means to write an equivalent expression which contains no similar terms.
This means that we will rewrite the expression with the fewest terms possible.
Therefore,
[tex]\frac{\frac{3x^{6} }{14y^{9} } }{\frac{33x^{4} }{10y^{2} } }[/tex]
The expression can be represented as follows:
3x⁶ / 14y⁹ ÷ 33x⁴ / 10y²
3x⁶ / 14y⁹ × 10y² / 33x⁴
Hence,
3x⁶ / 14y⁹ × 10y² / 33x⁴ = 30x⁶y² / 462x⁴y⁹
Therefore,
30x⁶y² / 462x⁴y⁹ = 10x² / 154y⁷ = 5x² / 77y⁷
Hence,
[tex]\frac{\frac{3x^{6} }{14y^{9} } }{\frac{33x^{4} }{10y^{2} } } = \frac{5x^{2} }{77y^{7} }[/tex]
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12.Work backwards to write a quadratic equation that will have solutions of x = -1/2 and x = 4. (Your equation must only have integer coefficients, meaning no fractions or decimals.)
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In general, a quadratic equation can be written in terms of its solutions:
[tex]y=(x-a)(x-b).[/tex]Now, notice that:
[tex]x+\frac{1}{2}=0\text{ }[/tex]when x= -1/2, and it is equivalent to:
[tex]2x+1=0.[/tex]Therefore, you can write the quadratic equation as:
[tex]y=(2x+1)(x-4).[/tex]Computing the above multiplication, you get:
[tex]y=2x^2-8x+x-4.[/tex]Simplifying the above equation you get:
[tex]y=2x^2-7x-4.[/tex]Answer: [tex]y=2x^{2}-7x-4[/tex]can you please solve this for me I'll make sure to give the best review
Answers
-9 is an integer
the location of -9 is with 41
6.3 is a repeating decimal
the location is with 5.86666...
-4/5 is a fraction
the location is with 11/12
I can't find the coordinates of midpoint D , must simplify
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We have to find the midpoint coordinates (D) of segment AB.
We can calculate the coordinates of the midpoint as:
[tex]\begin{gathered} x_M=\frac{x_A+x_B}{2}=\frac{-6x+(-2x)}{2}=\frac{-8x}{2}=-4x \\ \\ y_M=\frac{y_A+y_B}{2}=\frac{4y+(-4y)}{2}=\frac{4y-4y}{2}=0 \end{gathered}[/tex]Answer: D = (-4x, 0)
The formula k=5/9(f-32)+273.15 converts temperature of the object in a laboratory is cooled to 1.5 kelvin. What is the temperature of the object in degrees fahrenheit?
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The temperature of the object is -456.97 degrees
Fastex Shoes claims that it has designed a new range of lightweight shoes that allow athletes to run faster than they can using any other shoes.SportsPlus, a rival shoe manufacturer, then came up with a design that it claims performs even better than Fastex's design. A sports magazinedecided to test the claims. It created two groups of 15 randomly selected athletes each. Members of both groups were pretested anddetermined to have equal athletic ability. Additionally, they followed the same nutrition plan and training program during the study.The athletes did a 1-mile run to test the shoes. The magazine found that the athletes who used Fastex shoes had reduced their 1-mile run timeby an average of 396, and athletes who used Sports Plus shoes had reduced their 1-mile run time by an average of 2%.Given the magazine's data collection method and findings, what conclusion can be made?OA. The data collection method used was fair since both shoe designs were tested on identical parameters.OB. The data collection method used was nonrandom since the athletes using Fastex shoes may have trained better.OC. The data collection method used was unfair since the athletes given Fastex shoes may have been better runners than theathletes given SportsPlus shoes.OD. The data collection method used was random since the athletes in the two groups used two different shoe designs.
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B: The data collection method used was nonrandom since the athletes using Fastex shoes may have trained better.
This is false because they followed the same nutrition plan and training program during the study.
C. The data collection method used was unfair since the athletes given Fastex shoes may have been better runners than the athletes given SportsPlus shoes.
This is also false because members of both groups were pretested and determined to have equal athletic ability
D. Each group used one shoe design.
So, the correct option is A
Given f(x), find g(x) and h(x) such that f(x)= g(h(x)) and neither g(x) nor h(x) is solely x.
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Given:
[tex]\begin{gathered} f(x)=g(h(x)) \\ f(x)=\sqrt[]{-4x^2-3}+2 \end{gathered}[/tex]Solve :
[tex]g(h(x)=\sqrt[]{-4x^2-3}+2[/tex]The function g(x) convert then x is equal to h(x) then:
[tex]\begin{gathered} h(x)=-4x^2 \\ g(x)=\sqrt[]{x-3}+2 \end{gathered}[/tex]As a town gets smaller, the population of its high school decreases 6% each year. The senior class has 320 students now. In how many years will the high school have 100 students?
Answers
From the details provided, we know that the population of the town gets smaller, that is, a decline and not a growth. The annual rate of decline (or decay) is 6% (or 0.06). The formula for this is given as shown below;
[tex]y=a(1-r)^x_{}[/tex]The variables here are;
[tex]\begin{gathered} a=\text{initial value} \\ r=\text{rate of decline} \\ x=\text{period (in years)} \end{gathered}[/tex]The equation to represent the decline of this town's student population shall be;
[tex]\begin{gathered} y=320(1-0.06)^n \\ Simplified,\text{ we now have;} \\ y=320(0.94)^n \end{gathered}[/tex]When the population ofnthe town becomes 100, then we can replace variable y with 100. Since the formula is used to find the current population, and we have been given the population after a certain number of years, then our y is now 100.
We can now determine the number of years (variable n) that it takes before the population declines to 100 as shown below;
then our y is now 100.
We can now
A person standing 306 feet from the base of a church observed the angle of elevation to the church’s steeple to be 20°. How tall is the church. Give answer to the nearest whole number
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Solution
- The solution steps are given below:
[tex]\begin{gathered} \text{ Applying SOHCAHTOA, we have:} \\ \frac{h}{306}=\tan20 \\ h=306\tan20 \\ \\ h=111.374...ft\approx111ft\text{ \lparen To the nearest whole number\rparen} \end{gathered}[/tex]Final Answer
111 ft
Simplify the numerical expression (3^2 * 5^-1)^2
Answers
Simplify the numerical expression
[tex](3^2\cdot5^{-1})^{2}[/tex][tex]\begin{gathered} (9\cdot\frac{1}{5})^{2}= \\ (\frac{9}{5})^{2}= \\ \frac{81}{25} \end{gathered}[/tex]and In an experiment, the probability that event A occurs is 3/5 the probability that event B occurs is 3/4 and the probability that events A and B occur is 3/7what is the probability that A occurs given that B occurs
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Find the measure of the arc or central angle indicated. Assume that lines which appear to bediameters are actual diameters.
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From the given circle, the measure of the arc or the central angle indicated is as shown at the center of the circle is subtended by the arc
Hence, the measure of the arc or central angle indicated is 65° ,Option B
which answer choice gives the correct surface area for a triangular prism with bases that are 4 cm2 and sides that are 10 cm2? A. 12 cm2 B. 26 cm2 C.38 cm2 D. 40 cm2
Answers
Explanation
A trinagular prism has two bass and theree side surfaces.
Therefore, the suface area pf the prism is
[tex]S.A=3(10)+2(4)=30+8=38cm^2[/tex]Answer: Option C
Classify each set of measures as AAS, ASA, SSA, SAS, or SSS. Then find the indicated length or angle measure (to the nearest tenth).
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Hello there. To solve this question, we'll have to remember some properties about triangles.
Given the triangle:
Notice in this case we have two consecutive angles and a side between them. This is a case of ASA (angle-side-angle).
With respect to the side with measure x, we have two consecutive angles then the side, hence AAS.
To find x, we'll have to apply the law of sines:
[tex]\dfrac{A}{\sin(\alpha)}=\dfrac{B}{\sin(\beta)}=\dfrac{C}{\sin(\gamma)}=2R[/tex]In this case, the angle opposite to x measures 73º and the angle opposite to 4 measures 85º, hence:
[tex]\dfrac{x}{\sin(73^{\circ})}=\dfrac{4}{\sin(85^{\circ})}[/tex]Multiply both sides by a factor of sin(73º)
[tex]x=\dfrac{4\sin(73^{\circ})}{\sin(85^{\circ})}[/tex]Using a calculator, we get the following approximation (rounding to the nearest tenth):
[tex]x\approx3.8[/tex]This is the measure of x we're looking for.
In Mrs. Franco‘s class for every 64 is there a April right the ratio of boys to girls in simplest form
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The ratio of boys to girls in Mrs. Franco's class is 3:2 .
The Ratio is defined as the comparison of two quantities that have the same units .
In the question ,
it is given that
In Mrs. Franco's class
For every 6 boys there are 4 girls in the class
we have to find the ratio of , boys to girls
the number of boys = 6
the number of girls = 4
So , the ratio can be written as
boys / girls = 6/4
writing the ratio in the simplest form , we get
boys/girls = 3/2
the ratio is 3:2 .
Therefore , The ratio of boys to girls in Mrs. Franco's class is 3:2 .
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Rewrite the function by completing the square. f (x)= x^2 - 9x + 14
f (x) = _ ( x + _ )^2 + _
Answers
Answer:
f(x) = 1(x - 4.5)² - 6.25
Step-by-step explanation:
Hello!
Let's find the Vertex Form of the quadratic by Completing the Square.
f(x) = x² - 9x + 14x² - 9x + 14 = 0x² - 9x = -14The formula for a Perfect Square Trinomial is (a+b)² = a² + 2ab + b².
To find b², we need to divide -9 by 2 and square it.
-9-4.520.25Add this number to both sides and factor. Remember, the b term here is simply half of the b term in the equation (-4.5).
x² - 9x + 20.25 = -14 + 20.25(x - 4.5)² = 6.25(x - 4.5)²- 6.25 = 0Convert this back to function form:
f(x) = 1(x - 4.5)² - 6.25The equation is f(x) = 1(x - 4.5)² - 6.25.
Set up the equation for the following word problem and solve the equation. Let y be the unknown number.81 times a number minus 77 is equal to - 77 less than the number.Step 1 of 2: Write out the equation,
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The equation of the word is,
[tex]undefined[/tex]i need help, im confused
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Answer:
2
Step-by-step explanation:
So ABC and DEF are the same triangle, this question is asking me to write an equation between the relationships of DEF. How do I write that?
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Explanation
The first step is to draw a representation of the given parameters.
Since this is a right-angle triangle, the Pythagorean theorem applies. This can be seen below;
[tex]\text{Longest Leg}^2=sum\text{ of the square of the short legs}[/tex]We can then apply it to the sides of the triangle.
DF is the longest side. Therefore,
Answer
[tex]DF^2=DE^2+EF^2[/tex]6. sin D - Ог F 25 ot E 7. cos F. 24 8. sin F Nodule 13
Answers
In the given triangle :
FD = 25, FE = 7, DE = 24
SinD
From the trignometric ratio of sin: It express as the ratio of measurement of the side opposite to the angle to the measurement of hypotenuse of triangle DEF,
So, the SinD is express as :
[tex]\begin{gathered} \sin D=\frac{Opposite\text{ side}}{Hypotenuse} \\ \sin D=\frac{FE}{DF} \\ \sin D=\frac{7}{25} \end{gathered}[/tex]sin D = 7/25
cos F
From the trignometric ratio of cos : It expresses as the ratio of measurement of the side adjacent to the angle and to the hypotenuse of the triangle
So, the Cos F is express as :
[tex]\begin{gathered} \cos F=\frac{Adjacent\text{ side}}{Hypotenuse} \\ \cos F=\frac{FE}{DF} \\ \cos F=\frac{7}{25} \end{gathered}[/tex]cos F = 7/25
Sin F
From the trignometric ratio of sin: It express as the ratio of measurement of the side opposite to the angle to the measurement of hypotenuse of triangle DEF,
so, Sin F is express as :
[tex]\begin{gathered} \sin F=\frac{Opposite\text{ side}}{Hypotenuse} \\ \sin F=\frac{DE}{DF} \\ \sin F=\frac{24}{25} \end{gathered}[/tex]sin F = 24/25
Answer :
sin D = 7/25
cos F = 7/25
sin F = 24/25
Solve the inequalities|4x + 5| + 2 > 10
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We have to solve this inequality:
[tex]\begin{gathered} |4x+5|+2>10 \\ |4x+5|>10-2 \\ |4x+5|>8 \end{gathered}[/tex]We now use the properties of the absolute value. We will have two boundaries: one corresponding to when 4x+5 is negative and the other is when 4x+5 is positive.
When 4x+5 is negative, the absolute value function will change the sign of the expression, so we will have:
[tex]\begin{gathered} -(4x+5)>8 \\ -4x-5>8 \\ -4x>8+5 \\ -4x>13 \\ x<\frac{13}{-4} \\ x<-3.25 \end{gathered}[/tex]The other interval will be defined when 4x+5 is positive. In this case, the absolute function does not change the sign and we get:
[tex]\begin{gathered} 4x+5>8 \\ 4x>8-5 \\ 4x>3 \\ x>\frac{3}{4} \\ x>0.75 \end{gathered}[/tex]Then, the solution set is the union of the intervals x < -3.25 and x > 0.75.
We can express the interval as (-∞, -3.25) ∪ (0.75, ∞).
Answer: (-∞, -3.25) ∪ (0.75, ∞)
Miles east of 100 80 60 40 20 1 2 3 4 5 6 7 8 9 10 Time (hours) Where were the two cars in relation to each other when they began traveling? O A. Car B was 5 miles east of car A. O B. Car B was 20 miles east of car A. O C. Car Awas 15 miles east of car B. D. Car A was 5 miles east of car B. < PREVIOUS
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Car B was 5 miles east of car A, Option A
I would like to know the answer to -y+9x=0
Answers
Given
-y+9x=0
Find
check if equation model direct variation
Explanation
Equations with direct variation has a general form of y=kx
Given Equation
-y+9x=0
y=9x
whick is in the form of y=kx
Hence this equation is in direct variation with k=9
Final Answer
This equation is in direct variation with k=9
A pianist plans to play 4 pieces at a recital from her repertoire of 25 pieces, and is carefully consideringwhich song to play first, second, etc. to create a good flow. How many different recital programs arepossible?
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Given 25 pieces of repertoire, if a pianist plans to play 4 pieces at a recital and is considering playing which song to play first, second, etc, the possible ways is,
[tex]^{25}P_4=\frac{25!}{(25-4)!}=\frac{25!}{21!}=303600\text{ possible recital programs}[/tex]Hence, the different recital programs possible is 303600
Please help me step by step
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The value of the function f(x) at x = 0 is found as -1.
What is meant by the term function?A function is described as the connection between such a set of inputs that each have one output. A function is a relationship between inputs in which each input is linked to exactly one output. Every function does have a domain and a codomain, as well as a range. In general, a function is denoted by f(x), where x would be the input. A function's general representation is y = f(x). In mathematics, a function is a special relationship between inputs (the domain) and outputs (the codomain), where each input has precisely one output and the output could be traced all the way back to its input.For the given question,
The graph of the function f(x) = -x² + 4x - 1 is given.
For finding the value of f(x) at x = 0, check the y-coordinate of the graph when x = 0.
Put x = 0 in the given function.
f(0) = -0² + 40 - 1
f(0) = - 1
Thus, the value of the function f(x) at x = 0 is found as -1.
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3: Select the correct equation for the given situation. Then, select the solution for that equation. Two research submarines start to rise vertically toward the ocean surface. The Tri-I sub is at 4,863 feet below sea level (or -4,863 feet) and is ascending 81.1 feet per minute. The Quad-II sub is at 3,645 feet below sea level (or -3,645 feet) and is ascending 76.9 feet per minute. If the ocean surface is at 0 feet, how many minutes (m) must elapse for the two submarines to reach the same depth? m = 290 minutes m = 145 minutes m = 53.8 minutes 4,863 - 76.9m = 3,645 - 81.1m O -4,863 + 81.1m = - 3,645 + 76.9m 0 - 4, 863 + 76.9m = -3, 645 + 81.1m – 4, 863 – 81.2m 2 – 3, 645 + 76.9m Om < 53.8 minutes
Answers
Unknown, the correct equation is:
- 4,863 + 81.1m = - 3,645 + 76.9m
And to solve for m, you use the standard form:
Like terms:
81.1m - 76.9m = -3,645 + 4,863
4.2m = 1,218
Answer:Nuts
Step-by-step explanation:
green on green
Find the Value of interval [0,2pie] such as that tan s= -radical3/3
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The values of s in the interval [0, 2π) such that tan s = -(√3)/3 are 5π/6 and 11π/6.
What is trigonometry and how is it assessed?
Simply put, trigonometric functions—also referred to as circular functions: are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the fundamental trigonometric functions. Numerous trigonometric identities and formulas indicate the relationship between the functions and aid in determining the triangle's angles.
The quadrants determine the values of the trigonometric functions.
Given, tan s = -(√3)/3 ⇒ tan s = -(√3)/(√3)² ⇒ tan s = (-1)/(√3)
Therefore, the simplified value of tangent of s is tan s = (-1)/(√3)
Again the interval of the function is [0, 2π), so only the second and fourth quadrants can contain the given value of tangent being negative.
For the value of s in second quadrant, we have:
tan s = (-1)/(√3) ⇒ tan s = tan (π - (π/6)) ⇒ tan s = tan (5π/6) ⇒ s = 5π/6
For the value of s in the fourth quadrant, we have:
tan s = (-1)/(√3) ⇒ tan s = tan (2π - (π/6)) ⇒ tan s = tan (11π/6) ⇒ s = 11π/6
Thus, the values of s in the interval [0, 2π) such that tan s = -(√3)/3 are 5π/6 and 11π/6.
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Find R on line segment NM that is 1/4 the distance from N(-3,-3) toM(2, 3).R(x, y) =
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The distance on the x-coordinate from N to M is:
distance = 2 - (-3) = 2 + 3 = 5
Because 2 is the x-coordinate of M and -3 is the x-coordinate of N
Then, 1/4 of the distance is:
1/4*distance = (1/4)*5 = 5/4 = 1.25
So, the x-coordinate of R is:
(x-coordinate of N) + (1/4*distance) = -3 + 1.25 = -7/4 = -1.75
At the same way, the distance on the y-coordinae from N to M is:
distance = 3 - (-3) = 3 + 3 = 6
Then, 1/4 of the distance is:
1/4*distance = (1/4)*6 = 6/4 = 1.5
So, the y-coordinate of R is:
(y-coordinate of N) + (1/4*distance) = -3 + 1.5 = -1.5
Answer: R(x, y) = (-1.75, -1.5)
Which is the upper left quadrant on the coordinate plane?A coordinate plane.Quadrant IQuadrant IIQuadrant IIIQuadrant IV
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The quadrants on the coordinate plane are the following:
then, we have that the upper left quadrant is quadrant II
how many inches are in 20 centimeters?
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We know that an inch is equivalent to 2.54 centimeters, then if we want to know how many inches are in a centimeters we do this:
[tex]a\times\frac{1inch}{2.54\operatorname{cm}}[/tex]In this case, we have 20 centimeters, then replacing a by 20 we find the equivalent inches to 20 like this:
[tex]20\text{cm}\times\frac{1inch}{2.54\operatorname{cm}}\approx7.87\text{inches}[/tex]