Chapter 10

Solve by using the Quadratic Formula. \(3 t(t-2)=2\)

Solve by using the Quadratic Formula. \(4 d^{2}-7 d+2=0\)

Solve by using the Quadratic Formula. \(\frac{1}{9} c^{2}+\frac{2}{3} c=3\)

Solve by using the Quadratic Formula. \(2 x^{2}+12 x-3=0\)

Solve by using the Quadratic Formula. \(16 y^{2}+8 y+1=0\)

Determine the number of solutions to each quadratic equation. (a) \(9 v^{2}-15 v+25=0\) \(100 w^{2}+60 w+9=0\) \(5 c^{2}+7 c-10=0\) \(15 d^{2}-4 d+8=0\)

Determine the number of solutions to each quadratic equation. (a) \(r^{2}+12 r+36=0\) \(8 t^{2}-11 t+5=0\) \(4 u^{2}-12 u+9=0\) \(3 v^{2}-5 v-1=0\)

In the following exercises, determine the number of solutions to each quadratic equation. a. \(25 p^{2}+10 p+1=0\) b.\(7 q^{2}-3 q-6=0\) c.\(7 y^{2}+2 y+8=0\) d.\(25 z^{2}-60 z+36=0\)

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve. (a) \(x^{2}-5 x-24=0\) (b)\((y+5)^{2}=12\) (c)\(14 m^{2}+3 m=11\)

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve. (a)\((8 v+3)^{2}=81\) (b)\(w^{2}-9 w-22=0\) (c)\(4 n^{2}-10=6\)

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve. (a) \(6 a^{2}+14=20\) (b) \(\left(x-\frac{1}{4}\right)^{2}=\frac{5}{16}\) (c) \(y^{2}-2 y=8\)

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve. (a) \(8 b^{2}+15 b=4\) (b) \(\frac{5}{9} v^{2}-\frac{2}{3} v=1\) (c) \(\left(w+\frac{4}{3}\right)^{2}=\frac{2}{9}\)

A flare is fired straight up from a ship at sea. Solve the equation \(16\left(t^{2}-13 t+40\right)=0\) for \(t,\) the number of seconds it will take for the flare to be at an altitude of 640 feet.

An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation \(\frac{1}{2} h^{2}+3 h=140\) for \(h,\) the height of the window.

Solve the equation \(x^{2}+10 x=200\) by completing the square (b) using the Quadratic Formula (c) Which method do you prefer? Why?

Solve the equation \(12 y^{2}+23 y=24\) (a) by completing the square (b) using the Quadratic Formula ( Which method do you prefer? Why?

The product of two consecutive odd numbers is 255 . Find the numbers.

The product of two consecutive even numbers is 624 . Find the numbers.

The product of two consecutive odd numbers is \(1023 .\) Find the numbers.

The product of two consecutive odd numbers is \(483 .\) Find the numbers.

The product of two consecutive even numbers is 528 . Find the numbers.

The width of a triangle is six more than twice the height. The area of the triangle is 88 square yards. Find the height and width of the triangle.

The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is three feet. Find the lengths of the three sides of the triangle. Round to the nearest tenth.

The hypotenuse of a right triangle is \(10 \mathrm{~cm}\) long. One of the triangle's legs is three times the length of the other leg. Find the lengths of the three sides of the triangle. Round to the nearest tenth.

A farmer plans to fence off sections of a rectangular corral. The diagonal distance from one corner of the corral to the opposite corner is five yards longer than the width of the corral. The length of the corral is three times the width. Find the length of the diagonal of the corral. Round to the nearest tenth.

The length of a rectangular driveway is five feet more than three times the width. The area is 350 square feet. Find the length and width of the driveway.

A rectangular lawn has area 140 square yards. Its width that is six less than twice the length. What are the length and width of the lawn?

A firework rocket is shot upward at a rate of \(640 \mathrm{ft} / \mathrm{sec}\). Use the \(\quad\) projectile formula \(h=-16 t^{2}+v_{0} t\) to determine when the height of the firework rocket will be 1200 feet.

An arrow is shot vertically upward at a rate of 220 feet per second. Use the projectile formula \(h=-16 t^{2}+v_{0} t\) to determine when height of the arrow will be 400 feet.

A bullet is fired straight up from a BB gun with initial velocity 1120 feet per second at an initial height of 8 feet. Use the formula \(h=-16 t^{2}+v_{0} t+8\) to determine how many seconds it will take for the bullet to hit the ground. (That is, when will \(h=0\) ?)

A city planner wants to build a bridge across a lake in a park. To find the length of the bridge, he makes a right triangle with one leg and the hypotenuse on land and the bridge as the other leg. The length of the hypotenuse is 340 feet and the leg is 160 feet. Find the length of the bridge.

Make up a problem involving the product of two consecutive odd integers. Start by choosing two consecutive odd integers. (a) What are your integers? (b) What is the product of your integers? (c) Solve the equation \(n(n+2)=p,\) where \(p\) is the product you found in part (b). (1) Did you get the numbers you started with?

Make up a problem involving the product of two consecutive even integers. Start by choosing two consecutive even integers. (a) What are your integers? (b) What is the product of your integers? (c) Solve the equation \(n(n+2)=p,\) where \(p\) is the product you found in part (b). (d) Did you get the numbers you started with?

Recognize the Graph of a Quadratic Equation in Two Variables. $$ y=x^{2}+3 $$

Recognize the Graph of a Quadratic Equation in Two Variables. $$ y=-x^{2}+1 $$

In the following exercises, determine if the parabola opens up or down. $$ y=-2 x^{2}-6 x-7 $$

In the following exercises, determine if the parabola opens up or down. $$ y=6 x^{2}+2 x+3 $$

In the following exercises, determine if the parabola opens up or down. $$ y=4 x^{2}+x-4 $$

In the following exercises, determine if the parabola opens up or down. $$ y=-9 x^{2}-24 x-16 $$

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=x^{2}+8 x-1 $$

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=x^{2}+10 x+25 $$

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=-x^{2}+2 x+5 $$

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=-2 x^{2}-8 x-3 $$

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=x^{2}+10 x-11 $$

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=-x^{2}+8 x-19 $$

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=x^{2}+6 x+13 $$

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=4 x^{2}-20 x+25 $$

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=-x^{2}-14 x-49 $$

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=x^{2}+6 x+5 $$

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=x^{2}+4 x-12 $$