10.5 Graphing Quadratic Equations in Two Variables

Evaluate $2 x^{2} + 4 x - 1$ when $x = −3$.
If you missed this problem, review the reference.

Evaluate $- \frac{b}{2 a}$ when $a = \frac{1}{3}$ and $b = \frac{5}{6}$.
If you missed this problem, review the reference.

Determine whether each parabola opens upward or downward:

ⓐ $y = 2 x^{2} + 5 x - 2$ ⓑ $y = −3 x^{2} - 4 x + 7$

Determine whether each parabola opens upward or downward:

ⓐ $y = −2 x^{2} - 2 x - 3$ ⓑ $y = 5 x^{2} - 2 x - 1$

For the parabola $y = 2 x^{2} - 8 x + 1$ find: ⓐ the axis of symmetry and ⓑ the vertex.

For the parabola $y = 2 x^{2} - 4 x - 3$ find: ⓐ the axis of symmetry and ⓑ the vertex.

Find the intercepts of the parabola $y = x^{2} + 2 x - 8 .$

Find the intercepts of the parabola $y = x^{2} - 4 x - 12 .$

Find the intercepts of the parabola $y = 3 x^{2} + 4 x + 4 .$

Find the intercepts of the parabola $y = x^{2} - 4 x - 5 .$

Find the intercepts of the parabola $y = − x^{2} - 12 x - 36 .$

Find the intercepts of the parabola $y = 9 x^{2} + 12 x + 4 .$

Find the maximum or minimum value of the quadratic equation $y = x^{2} - 8 x + 12$.

Find the maximum or minimum value of the quadratic equation $y = −4 x^{2} + 16 x - 11$.

The quadratic equation $h = −16 t^{2} + 128 t + 32$ is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth.

A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of $h = −16 t^{2} + 208 t$. When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth.

Determine if the parabola opens up or down.

$y = −2 x^{2} - 6 x - 7$

Determine if the parabola opens up or down.

$y = 4 x^{2} + x - 4$

Find ⓐ the axis of symmetry and ⓑ the vertex.

$y = x^{2} + 8 x - 1$

Find ⓐ the axis of symmetry and ⓑ the vertex.

$y = − x^{2} + 2 x + 5$

Find the x- and y-intercepts.

$y = x^{2} + 7 x + 6$

Find the x- and y-intercepts.

$y = − x^{2} + 8 x - 19$

Find the x- and y-intercepts.

$y = 4 x^{2} - 20 x + 25$

Find the maximum or minimum value.

$y = 2 x^{2} + x - 1$

Find the maximum or minimum value.

$y = x^{2} - 6 x + 15$

Find the maximum or minimum value.

$y = −9 x^{2} + 16$

Solve. Round answers to the nearest tenth.

An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic equation $h = −16 t^{2} + 168 t + 45$ to find how long it will take the arrow to reach its maximum height, and then find the maximum height.

Solve. Round answers to the nearest tenth.

A computer store owner estimates that by charging $x$ dollars each for a certain computer, he can sell $40 - x$ computers each week. The quadratic equation $R = − x^{2} + 40 x$ is used to find the revenue, $R$, received when the selling price of a computer is $x$. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

Solve. Round answers to the nearest tenth.

A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation $A = x \left(240 - 2 x\right)$ gives the area of the corral, $A$, for the length, $x ,$ of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the…

For the revenue model in the exercise and the exercise, explain what the x-intercepts mean to the computer store owner.

Solve using the Square Root Property.

$x^{2} = 100$

Solve using the Square Root Property.

$m^{2} - 40 = 0$

Solve using the Square Root Property.

$4 a^{2} = 100$

Solve using the Square Root Property.

$r^{2} + 32 = 0$

Solve using the Square Root Property.

$\frac{4}{3} v^{2} + 4 = 28$

Solve using the Square Root Property.

$5 c^{2} + 3 = 19$

Solve using the Square Root Property.

$\left(p - 5\right)^{2} + 3 = 19$

Solve using the Square Root Property.

$\left(u + 1\right)^{2} = 45$

Solve using the Square Root Property.

$\left(x - \frac{1}{4}\right)^{2} = \frac{3}{16}$

Solve using the Square Root Property.

$\left(m - 7\right)^{2} + 6 = 30$

Solve using the Square Root Property.

$\left(5 c + 3\right)^{2} = −20$

Solve using the Square Root Property.

$m^{2} - 6 m + 9 = 48$

Solve using the Square Root Property.

$64 a^{2} + 48 a + 9 = 81$

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$x^{2} + 22 x$

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$m^{2} - 8 m$

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$a^{2} - 3 a$

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$p^{2} + \frac{4}{5} p$

Solve by completing the square.

$c^{2} + 20 c = 21$

Solve by completing the square.

$x^{2} - 4 x = 32$

Solve by completing the square.

$r^{2} + 6 r = −100$

Solve by completing the square.

$v^{2} - 14 v = −31$

Solve by completing the square.

$m^{2} + 10 m - 4 = −13$

Solve by completing the square.

$a^{2} = 3 a + 8$

Solve by completing the square.

$\left(u + 8\right) \left(u + 4\right) = 14$

Solve by completing the square.

$3 p^{2} - 18 p + 15 = 15$

Solve by completing the square.

$4 y^{2} - 6 y = 4$

Solve by completing the square.

$3 c^{2} + 2 c = 9$

Solve by using the Quadratic Formula.

$4 x^{2} - 5 x + 1 = 0$

Solve by using the Quadratic Formula.

$r^{2} - r - 42 = 0$

Solve by using the Quadratic Formula.

$4 v^{2} + v - 5 = 0$

Solve by using the Quadratic Formula.

$3 m^{2} + 8 m + 2 = 0$

Solve by using the Quadratic Formula.

$6 a^{2} - 5 a + 2 = 0$

Solve by using the Quadratic Formula.

$u \left(u - 10\right) + 3 = 0$

Solve by using the Quadratic Formula.

$\frac{1}{8} p^{2} - \frac{1}{5} p = - \frac{1}{20}$

Solve by using the Quadratic Formula.

$4 c^{2} + 4 c + 1 = 0$

Determine the number of solutions to each quadratic equation.

1. ⓐ $9 x^{2} - 6 x + 1 = 0$
2. ⓑ $3 y^{2} - 8 y + 1 = 0$
3. ⓒ $7 m^{2} + 12 m + 4 = 0$
4. ⓓ $5 n^{2} - n + 1 = 0$

Identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.

1. ⓐ $16 r^{2} - 8 r + 1 = 0$
2. ⓑ $5 t^{2} - 8 t + 3 = 9$
3. ⓒ $3 \left(c + 2\right)^{2} = 15$

Solve by using methods of factoring, the square root principle, or the quadratic formula.

Find two consecutive odd numbers whose product is 323.

Solve by using methods of factoring, the square root principle, or the quadratic formula.

A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.

Solve by using methods of factoring, the square root principle, or the quadratic formula.

The front walk from the street to Pam’s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.

Solve by using methods of factoring, the square root principle, or the quadratic formula.

A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula $h = −16 t^{2} + v_{0} t$ to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.

Determine if the following parabolas open up or down.

$y = −3 x^{2} + 3 x - 1$

Determine if the following parabolas open up or down.

$y = x^{2} + 8 x - 1$

Find ⓐ the axis of symmetry and ⓑ the vertex.

$y = − x^{2} + 6 x + 8$

Find the x- and y-intercepts.

$y = x^{2} - 4 x - 5$

Find the x- and y-intercepts.

$y = x^{2} - 4 x + 10$

Find the x- and y-intercepts.

$y = 16 x^{2} - 8 x + 1$

Find the minimum or maximum value.

$y = 7 x^{2} + 14 x + 6$

Solve. Rounding answers to the nearest tenth.

A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation $h = −16 t^{2} + 112 t$ to find how long it will take the ball to reach maximum height, and then find the maximum height.

Use the Square Root Property to solve the quadratic equation: $3 \left(w + 5\right)^{2} = 27$.

Use the Quadratic Formula to solve the quadratic equation: $2 m^{2} - 5 m + 3 = 0$.

Solve the following quadratic equations. Use any method.

$3 n^{2} + 8 n + 3 = 0$

Solve the following quadratic equations. Use any method.

$x \left(x + 3\right) + 12 = 0$

Use the discriminant to determine the number of solutions of each quadratic equation.

$6 p^{2} - 13 p + 7 = 0$

Solve by factoring, the Square Root Property, or the Quadratic Formula.

Find two consecutive even numbers whose product is 360.

For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and ⓔ the maximum or minimum value.

$y = 3 x^{2} + 6 x + 8$

For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and ⓔ the maximum or minimum value.

$y = x^{2} + 10 x + 24$

For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and ⓔ the maximum or minimum value.

$y = − x^{2} - 8 x + 16$