Chapter 5

Graph a quadratic equation that has a a. positive discriminant. b. negative discriminant. c. zero discriminant.

Compare the graphs of \(y=2(x-5)^{2}+4\) and \(y=2(x-4)^{2}-1\)

Find the values of \(m\) and \(n\) that make each equation true. $$ (4+n)+(3 m-7) i=8-2 i $$

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}+\frac{5}{2} x+c\)

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. $$ f(x)=-\frac{1}{2} x^{2}-2 x+3 $$

Explain why the roots of a quadratic equation are complex if the value of the discriminant is less than 0.

ACT/SAT If \((x+1)(x-2)\) is positive, which statement must be true? A \(x < -1\) or \(x > 2\) B \(x > -1\) or \(x < 2\) C \(-1 < x < 2\) D \(-2 < x < 1\)

AEROSPACE NASA's KC135 A aircraft flies in parabolic arcs to simulate the weightlessness experienced by astronauts in space. The height h of the aircraft (in feet) \(t\) seconds after it begins its parabolic flight can be modeled by the equation \(h(t)=-9.09(t-32.5)^{2}+34,000 .\) What is the maximum height of the aircraft during this maneuver and when does it occur?

ELECTRICITY. For Exercises 50 and \(51,\) use the formula \(E=I \cdot Z\) The current in a circuit is \(2+5 j\) amps, and the impedance is \(4-j\) ohms. What is the voltage?

Solve each equation by completing the square. \(x^{2}+1.4 x=1.2\)

ACT/SAT If one of the roots of the equation \(x^{2}+k x-12=0\) is \(4,\) what is the value of \(k ?\) \(\mathrm{A}-1\) \(\mathrm{B} 0\) \(\mathrm{C} 1\) \(\mathrm{D} 3\)

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. $$ f(x)=\frac{3}{4} x^{2}-5 x-2 $$

Find the exact solutions of \(2 i x^{2}-3 i x-5 i=0\) by using the Quadratic Formula.

DIVING For Exercises \(49-51\) , use the following information. The distance of a diver above the water \(d(t)\) (in feet) \(t\) seconds after diving off a platform is modeled by the equation \(d(t)=-16 t^{2}+8 t+30\) Find the time it will take for the diver to hit the water.

Solve each equation by completing the square. \(x^{2}-4.7 x=-2.8\)

Choose two integers. Then write an equation with those roots in standard form. How would the equation change if the signs of the two roots were switched?

Given the equation \(x^{2}+3 x-4=0\), a. Find the exact solutions by using the Quadratic Formula. b. Graph \(f(x)=x^{2}+3 x-4\) c. Explain how solving with the Quadratic Formula can help graph a quadratic function.

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=x^{2}-2 x+9 $$

DIVING For Exercises \(49-51\) , use the following information. The distance of a diver above the water \(d(t)\) (in feet) \(t\) seconds after diving off a platform is modeled by the equation \(d(t)=-16 t^{2}+8 t+30\) Write an equation that models the diver's distance above the water if the platform were 20 feet higher.

Find the sum of \(i x^{2}-(2+3 i) x+2\) and \(4 x^{2}+(5+2 i) x-4 i\)

Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x-\) -coordinate of the vertex for each quadratic function. Then graph the function by making a table of values. $$ f(x)=x^{2}-6 x+4 $$

Solve each equation by completing the square. \(x^{2}-\frac{2}{3} x-\frac{26}{9}=0\)

For a quadratic equation of the form \((x-p)(x-q)=0\) , show that the axis of symmetry of the related quadratic function is located halfway between the \(x\) -intercepts \(p\) and \(q .\)

CONSTRUCTION. For Exercises \(51-54,\) use the following information. Jaime has 120 feet of fence to make a rectangular kennel for his dogs. He will use his house as one side. What are reasonable values for the domain of the area function?

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=-2 x^{2}+16 x-32 $$

DIVING For Exercises \(49-51\) , use the following information. The distance of a diver above the water \(d(t)\) (in feet) \(t\) seconds after diving off a platform is modeled by the equation \(d(t)=-16 t^{2}+8 t+30\) Find the time it would take for the diver to hit the water from this new height.

Simplify \(\left[(3+i) x^{2}-i x+4+i\right]-\left[(-2+3 i) x^{2}+(1-2 i) x-3\right]\)

Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x-\) -coordinate of the vertex for each quadratic function. Then graph the function by making a table of values. $$ f(x)=-4 x^{2}+8 x-1 $$

Solve each equation by completing the square. \(x^{2}-\frac{3}{2} x-\frac{23}{16}=0\)

CONSTRUCTION. For Exercises \(51-54,\) use the following information. Jaime has 120 feet of fence to make a rectangular kennel for his dogs. He will use his house as one side. What dimensions produce a kennel with the greatest area?

If \(2 x^{2}-5 x-9=0,\) then \(x\) could be approximately equal to which of the following? A. \(-1.12\) B. 1.54 C. 2.63 D. 3.71

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=\frac{1}{2} x^{2}+6 x+18 $$

OPEN ENDED Write the equation of a parabola with a vertex of \((2,-1)\) and which opens downward.

Simplify. $$ \sqrt{-13} \cdot \sqrt{-26} $$

Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x-\) -coordinate of the vertex for each quadratic function. Then graph the function by making a table of values. $$ f(x)=\frac{1}{4} x^{2}+3 x+4 $$

Solve each equation by completing the square. \(3 x^{2}-4 x=2\)

Which quadratic equation has roots \(\frac{1}{2}\) and \(\frac{1}{3} ?\) A. \(5 x^{2}-5 x-2=0\) B. \(5 x^{2}-5 x+1=0\) C. \(6 x^{2}+5 x-1=0\) D. \(6 x^{2}-5 x+1=0\)

CONSTRUCTION. For Exercises \(51-54,\) use the following information. Jaime has 120 feet of fence to make a rectangular kennel for his dogs. He will use his house as one side. Find the maximum area of the kennel.

What are the \(x\) -intercepts of the graph of \(y=-2 x^{2}-5 x+12 ?\) F. \(-\frac{3}{2}, 4\) G. \(-4, \frac{3}{2}\) H. \(-2, \frac{1}{2}\) J. \(-\frac{1}{2}, 2\)

Solve each equation by using the method of your choice. Find exact solutions. $$ x^{2}+12 x+32=0 $$

CHALLENGE Given \(y=a x^{2}+b x+c\) with \(a \neq 0\) , derive the equation for the axis of symmetry by completing the square and rewriting the equation in the form \(y=a(x-h)^{2}+k\)

Simplify. $$ (4 i)\left(\frac{1}{2} i\right)^{2}(-2 i)^{2} $$

Solve the system \(4 x-y=0,2 x+3 y=14\) by using inverse matrices.

Solve each equation by completing the square. \(2 x^{2}-7 x=-12\)

What is the solution set for the equation \(3(4 x+1)^{2}=48 ?\) F. \(\left\\{\frac{5}{4},-\frac{3}{4}\right\\}\) G. \(\left\\{-\frac{5}{4}, \frac{3}{4}\right\\}\) H. \(\left\\{\frac{15}{4},-\frac{17}{4}\right\\}\) J. \(\left\\{\frac{1}{3},-\frac{4}{3}\right\\}\)

Solve each equation by using the Square Root Property. \(x^{2}+18 x+81=25\)

Solve each equation by using the method of your choice. Find exact solutions. $$ x^{2}+7=-5 x $$

Simplify. $$ i^{38} $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rr}{6} & {4} \\ {-3} & {2}\end{array}\right] $$

In an engineering test, a rocket sled is propelled into a target. The sled's distance \(d\) in meters from the target is given by the formula \(d=-1.5 t^{2}+120,\) where \(t\) is the number of seconds after rocket ignition. How many seconds have passed since rocket ignition when the sled is 10 meters from the target?