Chapter 10

COMMON FACTOR Factor the expression. $$ 4 b^{2}-40 b+100 $$

Use a vertical format or a horizontal format to add or subtract. $$ \left(3 x+2 x^{2}-4\right)-\left(x^{2}+x-6\right) $$

Solve \(x^{2}-9 x=36\) by factoring f. 12 and \(-3\) g. 12 and \(-3\) h. 4 and \(-9\) j. 9 and \(-4\)

Solve the equation. Tell which method you used. \(27+6 w-w^{2}=0\)

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (a+2 b)^{2}=a^{2}+2 a b+4 b^{2} $$

The cross section of the telescope’s dish can be modeled by the polynomial function $$y=\frac{14}{41^{2}}(x+41)(x-41)$$ where \(x\) and \(y\) are measured in feet, and the center of the dish is at \(x=0\) Use the model to find the coordinates of the center of the dish.

$$ (9 w-5)(7 w-12) $$

Solve the equation by factoring. $$ 2 x^{2}+19 x=-24 $$

COMMON FACTOR Factor the expression. $$ 32 x^{2}-48 x+18 $$

Use a vertical format or a horizontal format to add or subtract. $$ \left(u^{3}-u\right)-\left(u^{2}+5\right) $$

The length of a rectangular plot of land is 24 meters more than its width. A paved area measuring 8 meters by 12 meters is placed on the plot. The area of the unpaved part of the land is then 880 square meters. If w represents the width of the plot of land in meters, which of the following equations can be factored to find the possible values of w? HINT: Begin by drawing and labeling a diagram. $$ a.\quad w^{2}+24 w=880 $$ $$ b.\quad w^{2}+24 w+96=880 $$ $$ c.\quad w^{2}+24 w-96=880 $$ $$ d.\quad w^{2}+24 w=96 $$

Solve the equation. Tell which method you used. \(5 x^{4}-80 x^{2}=0\)

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (3 s+2 t)(3 s-2 t)=9 s^{2}+4 t^{2} $$

Use a vertical format to find the product. $$ (x+2)\left(x^{2}+3 x+5\right) $$

The Gateway Arch in St. Louis, Missouri, has the shape of a catenary (a U-shaped curve similar to a parabola). It can be approximated by the following model, where x and y are measured in feet. Gateway Arch model: \(y=-\frac{7}{1000}(x+300)(x-300)\) How far apart are the legs of the arch at the base?

Solve the equation by factoring. $$ 4 x^{2}-8 x=-3 $$

COMMON FACTOR Factor the expression. $$ 16 w^{2}+80 w+100 $$

Use a vertical format or a horizontal format to add or subtract. $$ \left(-7 x^{2}+12\right)-\left(6-4 x^{2}\right) $$

A triangle’s base is 16 feet less than 2 times its height. If h represents the height in feet, and the total area of the triangle is 48 square feet, which of the following equations can be used to determine the height? $$ f.\quad 2 h+2(h+4)=48 $$ $$ g.\quad h^{2}-8 h=48 $$ $$ h.\quad h^{2}+8 h=48 $$ $$ j.\quad 2 h^{2}-16 h=48 $$

Solve the equation. Tell which method you used. \(-16 x^{3}+4 x=0\)

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (9 x+8)(9 x-8)=81 x^{2}-64 $$

$$ (d-5)\left(d^{2}-2 d-6\right) $$

Solve the equation by factoring. $$ 6 x^{2}-23 x=18 $$

COMMON FACTOR Factor the expression. $$ 2 x^{2}+28 x y+98 y^{2} $$

Use a vertical format or a horizontal format to add or subtract. $$ \left(10 x^{3}+2 x^{2}-11\right)+\left(9 x^{2}+2 x-1\right) $$

Find the greatest common factor. $$ 12,36 $$

Solve the equation. Tell which method you used. \(10 x^{3}-290 x^{2}-620 x=0\)

$$ (a-3)\left(a^{2}-4 a-6\right) $$

Use the following equation which models a cross section of the Barringer Meteor Crater, near Winslow, Arizona. Note that x and y are measured in meters and the center of the crater is at x 0. Barringer Meteor model: \(y=\frac{1}{1800}(x-600)(x+600)\) Assuming the lip of the crater is at y 0, how wide is the crater?

Solve the equation by factoring. $$ 8 x^{2}-34 x+24=-11 $$

Solve the equation by factoring. Use a graphing calculator to check your solution if you wish. $$ 4 x^{2}+4 x+1=0 $$

In Exercises 51 and \(52,\) use the following information. You plan to build a house that is 1.5 times as long as it is wide. You want the land around the house to be 20 feet wider than the width of the house, and twice as long as the length of the house, as shown in the figure below. Write an expression for the area of the land surrounding the house.

Find the greatest common factor. $$ 30,45 $$

Use the quadratic formula or factoring to find the roots of the polynomial. Write your solutions in simplest form. \(4 x^{2}-9 x-9=0\)

$$ (2 x+3)\left(3 x^{2}-4 x+2\right) $$

Solve the equation by factoring. $$ 6 x^{2}+19 x-10=-20 $$

Solve the equation by factoring. Use a graphing calculator to check your solution if you wish. $$ 25 x^{2}-4=0 $$

Find the greatest common factor. $$ 24,72 $$

Use the quadratic formula or factoring to find the roots of the polynomial. Write your solutions in simplest form. \(5 x^{2}+2 x-3=0\)

Use a horizontal format to find the product. $$ (x+4)\left(x^{2}-2 x+3\right) $$

Solve \(6(x-3)(x+5)(x-9)=0\). A) \(6,3,5,\) and 9 B) \(3,-5,\) and 9 C) \(6,3,-5,\) and 9 D) \(6,3,5,\) and \(-9\)

Solve the equation by factoring. $$ 28 x^{2}-9 x-1=-4 x+2 $$

Solve the equation by factoring. Use a graphing calculator to check your solution if you wish. $$ 3 x^{2}-24 x+48=0 $$

In Exercises 53 and \(54,\) use the following information. From 1989 through \(1993,\) the amounts (in billions of dollars) spent on natural gas \(N\) and electricity \(E\) by United States residents can be modeled by the following equations, where \(t\) is the number of years since \(1989 .\) $$\text {Gas spending model:} N=1.488 t^{2}-3.403 t+65.590$$ $$\text {Electricity spending model:} E=-0.107 t^{2}+6.897 t+169.735$$ Find a model for the total amount \(A\) (in billions of dollars) spent on natural gas and electricity by United States residents from 1989 through \(1993 .\)

Find the greatest common factor. $$ 49,64 $$

Use the quadratic formula or factoring to find the roots of the polynomial. Write your solutions in simplest form. \(2 x^{2}+5 x+1=0\)

$$ (a-2)\left(a^{2}+6 a-7\right) $$

Solve the equation by factoring. $$ 10 x^{2}+x-10=-2 x+8 $$

Solve the equation by factoring. Use a graphing calculator to check your solution if you wish. $$ -27+3 x^{2}=0 $$

In Exercises 53 and \(54,\) use the following information. From 1989 through \(1993,\) the amounts (in billions of dollars) spent on natural gas \(N\) and electricity \(E\) by United States residents can be modeled by the following equations, where \(t\) is the number of years since \(1989 .\) $$\text {Gas spending model:} N=1.488 t^{2}-3.403 t+65.590$$ $$\text {Electricity spending model:} E=-0.107 t^{2}+6.897 t+169.735$$ According to the models, will more money be spent on natural gas or on electricity in \(2020 .\) HINT: It may be helpful to graph the equations on a graphing calculator to answer this question.