Chapter 5

Solve each inequality using a graph, a table, or algebraically. $$ x^{2}<10 x-25 $$

Find the exact solutions by using the Quadratic Formula. \(4 x^{2}+20 x+25=-2\)

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-12 x+c\)

Solve each equation by factoring. Then graph. \(x^{2}+6 x-16=0\)

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ x^{2}+16 x+64=-6 $$

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=x^{2}-x-6 $$

Solve each inequality using a graph, a table, or algebraically. $$ x^{2} \leq 3 $$

The height \(h(t)\) in feet of an object \(t\) seconds after it is propelled straight up from the ground with an initial velocity of 85 feet per second is modeled by the equation \(h(t)=-16 t^{2}+85 t\). When will the object be at a height of 50 feet?

Simplify. $$ i^{29} $$

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-3 x+c\)

Solve each equation by factoring. Then graph. \(4 x^{2}-13 x=12\)

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ x^{2}-12 x=-37 $$

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=4 x^{2}+12 x+9 $$

BASEBALL A baseball player hits a high pop-up with an initial upward velocity of 30 meter: per second, 1.4 meters above the ground. The height \(h(t)\) of the ball in meters \(t\) seconds after being hit is modeled by \(h(t)=-4.9 t^{2}+30 t+\) 1.4. How long does a player on the opposing team have to get under the ball if he opposing 1.7 meters above the ground? Does your answer seem reasonable? Explain.

The height \(h(t)\) in feet of an object \(t\) seconds after it is propelled straight up from the ground with an initial velocity of 85 feet per second is modeled by the equation \(h(t)=-16 t^{2}+85 t\). Will the object ever reach a height of 120 feet? Explain your reasoning.

Solve each equation by completing the square. \(x^{2}+3 x-18=0\)

Simplify. $$ i^{80} $$

Solve each equation by factoring. Then graph. \(x^{2}-14 x=-49\)

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 4 x^{2}-7 x-15=0 $$

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=-x^{2}-4 x+1 $$

Graph each inequality. $$ y \geq x^{2}+3 x-18 $$

Fountains The height of a fountain's water stream can be modeled by a quadratic function. Suppose the water from a jet reaches a maximum height of 8 feet at a distance 1 foot away from the jet. If the water lands 3 feet away from the jet, find a quadratic function that models the height \(H(d)\) of the water at any given distance \(d\) feet from the jet. Then compare the graph of the function to the parent function.

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. Do your answers for Exercises 1, 3, 5, and 7 fit these descriptions, respectively? \(8 x^{2}+18 x-5=0\)

Solve each equation by completing the square. \(x^{2}-8 x+11=0\)

Solve each equation. $$ 2 x^{2}+18=0 $$

Solve each equation by factoring. Then graph. \(x^{2}+9=6 x\)

Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ 2 x^{2}-2 x-3=0 $$

Graph each inequality. $$ y<-x^{2}+7 x+8 $$

Fountains The height of a fountain's water stream can be modeled by a quadratic function. Suppose the water from a jet reaches a maximum height of 8 feet at a distance 1 foot away from the jet. Suppose a worker increases the water pressure so that the stream reaches a maximum height of 12.5 feet at a distance of 15 inches from the jet. The water now lands 3.75 feet from the jet. Write a new quadratic function for \(H(d) .\) How do the changes in \(h\) and \(k\) affect the shape of the graph?

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. Do your answers for Exercises 1, 3, 5, and 7 fit these descriptions, respectively? \(4 x^{2}+4 x+1=0\)

Solve each equation by completing the square. \(2 x^{2}-3 x-3=0\)

Solve each equation. $$ -5 x^{2}-25=0 $$

Solve each equation by factoring. Then graph. \(x^{2}-3 x=-\frac{9}{4}\)

NUMBER THEORY Use a quadratic equation to find two real numbers with a sum of 5 and a product of \(-14,\) or show that no such numbers exist.

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=2 x^{2} $$

Graph each inequality. $$ y \leq x^{2}+4 x+4 $$

Graph each function. $$ y=4(x+3)^{2}+1 $$

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. Do your answers for Exercises 1, 3, 5, and 7 fit these descriptions, respectively? \(2 x^{2}-4 x+1=0\)

Find the values of \(m\) and \(n\) that make each equation true. $$ 2 m+(3 n+1) i=6-8 i $$

Solve each equation by completing the square. \(3 x^{2}+12 x-18=0\)

ARCHERY An arrow is shot upward with a velocity of 64 feet per second. Ignoring the height of the archer, how long after the arrow is released does it hit the ground? Use the formula \(h(t)=v_{0} t-16 t^{2}\) where \(h(t)\) is the height of an object in feet, \(v_{0}\) is the object's initial velocity in feet per second, and \(t\) is the time in seconds.

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=-5 x^{2} $$

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. Do your answers for Exercises 1, 3, 5, and 7 fit these descriptions, respectively? \(x^{2}+3 x+8=5\)

Graph each inequality. $$ y \leq x^{2}+4 x $$

Graph each function. $$ y=-(x-5)^{2}-3 $$

Find the values of \(m\) and \(n\) that make each equation true. $$ (2 n-5)+(-m-2) i=3-7 i $$

Solve each equation by completing the square. \(x^{2}+2 x+6=0\)

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=x^{2}+4 $$

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(-12 x^{2}+5 x+2=0\)

Graph each inequality. $$ y>x^{2}-36 $$