Chapter 12

NUMBER OF X-INTERCEPTS Determine whether the graph of the function intersects the \(x\) -axis in zero, one, or two points. $$ y=x^{2}+8 x+12 $$

Factor the expression completely. $$ 2 t^{3}-98 t $$

Solve the quadratic equation. $$ x^{2}-6 x-10=0 $$

Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}-30=-3 $$

Simplify the radical expression. $$ \frac{\sqrt{3}}{\sqrt{3}-1} $$

Find the domain of the function. Then sketch its graph and find the range. $$y=\sqrt{x-4}$$

Solve the equation. Check for extraneous solutions. $$ 2 \sqrt{x}=-18 $$

NUMBER OF X-INTERCEPTS Determine whether the graph of the function intersects the \(x\) -axis in zero, one, or two points. $$ y=x^{2}+x-10 $$

Factor the expression completely. $$ 2 x^{4}-8 x^{2} $$

Solve the quadratic equation. $$ x^{2}-12 x-3=0 $$

Solve the equation. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ x^{2}=\frac{20}{25} $$

Find the domain of the function. Then sketch its graph and find the range. $$y=\sqrt{x+1}$$

Solve the equation. Check for extraneous solutions. $$ x=\sqrt{-x+12} $$

NUMBER OF X-INTERCEPTS Determine whether the graph of the function intersects the \(x\) -axis in zero, one, or two points. $$ y=x^{2}+8 x+16 $$

Factor the expression completely. $$ c^{3}+2 c^{2}-8 c-16 $$

Solve the quadratic equation. $$ x^{2}-18 x+5=0 $$

Solve the equation. $$ (x+4)^{2}=0 $$

Find the domain of the function. Then sketch its graph and find the range. $$y=\sqrt{3 x+1}$$

Solve the equation. Check for extraneous solutions. $$ 2 \sqrt{x}+7=19 $$

NUMBER OF X-INTERCEPTS Determine whether the graph of the function intersects the \(x\) -axis in zero, one, or two points. $$ y=x^{2}+3 x+1 $$

Simplify the expression. $$ \frac{4 x}{28} $$

Solve the quadratic equation. $$ x^{2}-2 x-4=0 $$

Solve the equation. $$ (x+4)(x-8)=0 $$

A pole-vaulter’s approach velocity v (in feet per second) and height reached h (in feet) are related by the following equation. $$ v=8 \sqrt{h} $$ Suppose you are a pole-vaulter and reach a height of 20 feet and your opponent reaches a height of 16 feet. Write an expression that shows how much faster you ran than your opponent. Simplify the expression and round your answer to the nearest hundredth.

Find the domain of the function. Then sketch its graph and find the range. $$y=2 \sqrt{4 x+10}$$

Use the following information. During the hammer throw event, a hammer is swung around in a circle several times until the thrower releases it. As the hammer travels in the path of the circle, it accelerates toward the center. This acceleration is known as centripetal acceleration. The speed \(s\) that the hammer is thrown can be modeled by the formula \(s=\sqrt{1.2 a},\) where \(a\) is the centripetal acceleration of the hammer prior to being released. Find the approximate centripetal acceleration (in meters per second per second) when the ball is thrown with a speed of 18 meters per second.

NUMBER OF X-INTERCEPTS Determine whether the graph of the function intersects the \(x\) -axis in zero, one, or two points. $$ y=x^{2}-8 x-11 $$

Simplify the expression. $$ \frac{15 x}{75} $$

Explain why the quadratic formula gives real solutions only if \(a \neq 0\) and \(b^{2}-4 a c \geq 0\).

Solve the equation. $$ x(x-14)^{2}=0 $$

Many birds drop clams or other shellfish in order to break the shell and get the food inside. The time t (in seconds) it takes such an object to fall a certain distance d (in feet) is given by the following equation. $$ t=\frac{\sqrt{d}}{4} $$ A gull drops a clam from a height of 50 feet. A second gull drops a clam from a height of 32 feet. Write an expression that shows the difference in the time that it takes for the two clams to reach the ground. Simplify the expression.

Use the following information. In a natural history museum you see leg bones for two species of dinosaurs and want to know how fast they walked. The maximum walking speed \(S\) (in fect per scond) of a dinosaur can be modeled by the equation below, where \(L\) is the length (in feet) of the dinosaur's leg. Walking speed model: \(S=\sqrt{32 L}\) Find the domain of the walking speed model. Then sketch its graph.

Use the following information. During the hammer throw event, a hammer is swung around in a circle several times until the thrower releases it. As the hammer travels in the path of the circle, it accelerates toward the center. This acceleration is known as centripetal acceleration. The speed \(s\) that the hammer is thrown can be modeled by the formula \(s=\sqrt{1.2 a},\) where \(a\) is the centripetal acceleration of the hammer prior to being released. Find the approximate centripetal acceleration (in meters per second per second) when the ball is thrown with a speed of 24 meters per second.

ESTIMATING AREA Estimate the area of a rectangle whose sides are given. First round each side length to the nearest whole number. Then multiply to find the area. $$ 5.1 \text { by } 7.2 $$

Simplify the expression. $$ \frac{-48 x^{3}}{-12 x^{2}} $$

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. In art class you are designing the floor plan of a house. The kitchen is supposed to have 150 square feet of space. What should the dimensions of the kitchen floor be if you want it to be square?

Determine whether the number is prime or composite. If it is composite, give its prime factorization. $$ 13 $$

Simplify \(\sqrt{5}(6+\sqrt{5})\) $$ a.\quad \sqrt{30}+5 $$ $$ b.\quad 5 \sqrt{6}+5 $$ $$ c.\quad 6 \sqrt{5}+5 $$ $$ d.\quad 11 \sqrt{5} $$

Use the following information. In a natural history museum you see leg bones for two species of dinosaurs and want to know how fast they walked. The maximum walking speed \(S\) (in fect per scond) of a dinosaur can be modeled by the equation below, where \(L\) is the length (in feet) of the dinosaur's leg. Walking speed model: \(S=\sqrt{32 L}\) For one dinosaur the length of the leg is 1 foot. For the other dinosaur the length of the leg is 4 feet. How much faster does the taller dinosaur walk than the shorter dinosaur?

ESTIMATING AREA Estimate the area of a rectangle whose sides are given. First round each side length to the nearest whole number. Then multiply to find the area. $$ 10.6 \text { by } 17.3 $$

Simplify the expression. $$ \frac{18 x^{3}}{56 x^{7}} $$

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. A rectangle is \(2 x\) feet long and \(x+5\) feet wide. The area is 600 square feet. What are the dimensions of the rectangle?

Which of the following is equal to the difference \(\sqrt{3}-5 \sqrt{9} ?\) $$ f.\quad \sqrt{3}-15 $$ $$ g.\quad -4 \sqrt{3} $$ $$ h.\quad \sqrt{3}-3 $$ $$ j.\quad 3+2 \sqrt{5} $$

Determine whether the number is prime or composite. If it is composite, give its prime factorization. $$ 28 $$

Which of the following is a solution of \(x=\sqrt{30-x} ?\) A. -6 B. 0 C. 5 D. 30

ESTIMATING AREA Estimate the area of a rectangle whose sides are given. First round each side length to the nearest whole number. Then multiply to find the area. $$ 5.1 \text { by } 9.9 $$

Simplify the expression. $$ \frac{-3 x^{2}+21 x}{12 x^{2}} $$

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. The base of a triangle is \(x\) feet and the height is \((4+2 x)\) feet. The area of the triangle is 60 square feet. What are the dimensions of the triangle?

Simplify \(\frac{3}{5-\sqrt{2}}\) $$ a.\quad \frac{15+3 \sqrt{2}}{23} $$ $$ b.\quad \frac{15+3 \sqrt{2}}{25} $$ $$ c.\quad\frac{15+\sqrt{6}}{23} $$ $$ d.\quad \frac{15+\sqrt{6}}{25} $$

Determine whether the number is prime or composite. If it is composite, give its prime factorization. $$ 75 $$