10.3 Solve Quadratic Equations Using the Quadratic Formula

Before you get started, take this readiness quiz.

Simplify: $\frac{−20 - 5}{10}$.
If you missed this problem, review the reference.

Simplify: $4 + \sqrt{121}$.
If you missed this problem, review the reference.

Simplify: $\sqrt{128}$.
If you missed this problem, review the reference.

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

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

Solve $a^{2} - 2 a - 15 = 0$ by using the Quadratic Formula.

Solve $b^{2} + 10 b + 24 = 0$ by using the Quadratic Formula.

Solve $2 p^{2} + 8 p + 5 = 0$ by using the Quadratic Formula.

Solve $5 q^{2} - 11 q + 3 = 0$ by using the Quadratic Formula.

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

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

Solve $4 a^{2} - 3 a + 8 = 0$ by using the Quadratic Formula.

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

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

Solve $y \left(3 y - 1\right) - 2 = 0$ by using the Quadratic Formula.

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

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

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

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

Determine the number of solutions to each quadratic equation:

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

Determine the number of solutions to each quadratic equation:

ⓐ $8 m^{2} - 3 m + 6 = 0$ ⓑ $5 z^{2} + 6 z - 2 = 0$ ⓒ $9 w^{2} + 24 w + 16 = 0$ ⓓ $9 u^{2} - 2 u + 4 = 0$

Determine the number of solutions to each quadratic equation:

ⓐ $b^{2} + 7 b - 13 = 0$ ⓑ $5 a^{2} - 6 a + 10 = 0$ ⓒ $4 r^{2} - 20 r + 25 = 0$ ⓓ $7 t^{2} - 11 t + 3 = 0$

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.

$r^{2} - 8 r - 33 = 0$

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.

$2 x^{2} + 3 x + 9 = 0$

Solve by using the Quadratic Formula.

$v \left(v + 5\right) - 10 = 0$

Solve by using the Quadratic Formula.

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

Solve by using the Quadratic Formula.

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

Solve by using the Quadratic Formula.

$5 m^{2} + 2 m - 7 = 0$

Solve by using the Quadratic Formula.

$p^{2} - 6 p - 27 = 0$

Solve by using the Quadratic Formula.

$4 r^{2} + 3 r - 5 = 0$

Solve by using the Quadratic Formula.

$2 a^{2} + 12 a + 5 = 0$

Solve by using the Quadratic Formula.

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

Solve by using the Quadratic Formula.

$2 x^{2} + 12 x - 3 = 0$

Determine the number of solutions to each quadratic equation.

1. ⓐ $4 x^{2} - 5 x + 16 = 0$
2. ⓑ $36 y^{2} + 36 y + 9 = 0$
3. ⓒ $6 m^{2} + 3 m - 5 = 0$
4. ⓓ $18 n^{2} - 7 n + 3 = 0$

Determine the number of solutions to each quadratic equation.

1. ⓐ $r^{2} + 12 r + 36 = 0$
2. ⓑ $8 t^{2} - 11 t + 5 = 0$
3. ⓒ $4 u^{2} - 12 u + 9 = 0$
4. ⓓ $3 v^{2} - 5 v - 1 = 0$

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.

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

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.

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