9.3 Solve Quadratic Equations Using the Quadratic Formula

Before you get started, take this readiness quiz.

Evaluate $b^{2} - 4 a b$ when $a = 3$ and $b = −2 .$
If you missed this problem, review the reference.

Simplify: $\sqrt{108} .$
If you missed this problem, review the reference.

Simplify: $\sqrt{50} .$
If you missed this problem, review the reference.

Solve by using the Quadratic Formula: $3 y^{2} - 5 y + 2 = 0$.

Solve by using the Quadratic Formula: $4 z^{2} + 2 z - 6 = 0$.

Solve by using the Quadratic Formula: $a^{2} - 2 a = 15$.

Solve by using the Quadratic Formula: $b^{2} + 24 = −10 b$.

Solve by using the Quadratic Formula: $3 m^{2} + 12 m + 7 = 0$.

Solve by using the Quadratic Formula: $5 n^{2} + 4 n - 4 = 0$.

Solve by using the Quadratic Formula: $4 a^{2} - 2 a + 8 = 0$.

Solve by using the Quadratic Formula: $5 b^{2} + 2 b + 4 = 0$.

Solve by using the Quadratic Formula: $x \left(x + 2\right) - 5 = 0 .$

Solve by using the Quadratic Formula: $3 y \left(y - 2\right) - 3 = 0 .$

Solve by using the Quadratic Formula: $\frac{1}{4} c^{2} - \frac{1}{3} c = \frac{1}{12}$.

Solve by using the Quadratic Formula: $\frac{1}{9} d^{2} - \frac{1}{2} d = - \frac{1}{3}$.

Solve by using the Quadratic Formula: $r^{2} + 10 r + 25 = 0 .$

Solve by using the Quadratic Formula: $25 t^{2} - 40 t = −16 .$

Determine the number of solutions to each quadratic equation.

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

Determine the numberand type of solutions to each quadratic equation.

ⓐ $8 m^{2} - 3 m + 6 = 0$ ⓑ $5 z^{2} + 6 z - 2 = 0$ ⓒ $9 w^{2} + 24 w + 16 = 0 .$

Determine the number and type of solutions to each quadratic equation.

ⓐ $b^{2} + 7 b - 13 = 0$ ⓑ $5 a^{2} - 6 a + 10 = 0$ ⓒ $4 r^{2} - 20 r + 25 = 0 .$

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ $5 z^{2} = 17$ ⓑ $4 x^{2} - 12 x + 9 = 0$ ⓒ $8 u^{2} + 6 u = 11 .$

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ $x^{2} + 6 x + 8 = 0$ ⓑ $\left(n - 3\right)^{2} = 16$ ⓒ $5 p^{2} - 6 p = 9 .$

Identify the most appropriate method to use to solve each quadratic equation.

ⓐ $8 a^{2} + 3 a - 9 = 0$ ⓑ $4 b^{2} + 4 b + 1 = 0$ ⓒ $5 c^{2} = 125 .$

Solve by using the Quadratic Formula.

$4 m^{2} + m - 3 = 0$

Solve by using the Quadratic Formula.

$2 p^{2} - 7 p + 3 = 0$

Solve by using the Quadratic Formula.

$p^{2} + 7 p + 12 = 0$

Solve by using the Quadratic Formula.

$r^{2} - 8 r = 33$

Solve by using the Quadratic Formula.

$3 u^{2} + 7 u - 2 = 0$

Solve by using the Quadratic Formula.

$2 a^{2} - 6 a + 3 = 0$

Solve by using the Quadratic Formula.

$x^{2} + 8 x - 4 = 0$

Solve by using the Quadratic Formula.

$3 y^{2} + 5 y - 2 = 0$

Solve by using the Quadratic Formula.

$2 x^{2} + 3 x + 3 = 0$

Solve by using the Quadratic Formula.

$8 x^{2} - 6 x + 2 = 0$

Solve by using the Quadratic Formula.

$\left(v + 1\right) \left(v - 5\right) - 4 = 0$

Solve by using the Quadratic Formula.

$\left(y + 4\right) \left(y - 7\right) = 18$

Solve by using the Quadratic Formula.

$\frac{1}{3} m^{2} + \frac{1}{12} m = \frac{1}{4}$

Solve by using the Quadratic Formula.

$\frac{3}{4} b^{2} + \frac{1}{2} b = \frac{3}{8}$

Solve by using the Quadratic Formula.

$16 c^{2} + 24 c + 9 = 0$

Solve by using the Quadratic Formula.

$25 q^{2} + 30 q + 9 = 0$

Determine the number of real solutions for each quadratic equation.

ⓐ $4 x^{2} - 5 x + 16 = 0$ ⓑ $36 y^{2} + 36 y + 9 = 0$ ⓒ $6 m^{2} + 3 m - 5 = 0$

Determine the number of real solutions for each quadratic equation.

ⓐ $r^{2} + 12 r + 36 = 0$ ⓑ $8 t^{2} - 11 t + 5 = 0$ ⓒ $3 v^{2} - 5 v - 1 = 0$

Identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

ⓐ $x^{2} - 5 x - 24 = 0$
ⓑ $\left(y + 5\right)^{2} = 12$
ⓒ $14 m^{2} + 3 m = 11$

Identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

ⓐ $6 a^{2} + 14 a = 20$
ⓑ $\left(x - \frac{1}{4}\right)^{2} = \frac{5}{16}$
ⓒ $y^{2} - 2 y = 8$

Solve the equation $x^{2} + 10 x = 120$

ⓐ by completing the square

ⓑ using the Quadratic Formula

ⓒ Which method do you prefer? Why?