# 10.5: Quadratic Equations by the Quadratic Formula

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Solve quadratic equations using the quadratic formula.
- Identify and choose methods for solving quadratic equations.
- Solve real-world problems using functions by completing the square.

## Vocabulary

Terms introduced in this lesson:

- quadratic formula
- roots, solutions to a quadratic equation

## Teaching Strategies and Tips

Emphasize that the quadratic formula comes from *completing the square* of a general quadratic equation.

- Point out that half the coefficient of \begin{align*}x\end{align*} squared can be found by
*multiplying*by \begin{align*}1/2\end{align*}:

\begin{align*}\left (\frac{b}{a} \cdot \frac{1}{2} \right )^2 = \left (\frac{b}{2a} \right )^2\end{align*}

- Remind students to find common denominators before simplifying the right side of the equation.

Have students rewrite each quadratic equation in standard form first. See Example 3.

Use Example 4 to show students how one goes about choosing a solving method.

Teachers are encouraged to use quadratic equations with non-integer coefficients—decimals and fractions—as additional practice problems.

## Error Troubleshooting

General Tip: Some students may use the values of \begin{align*}a\end{align*}, \begin{align*}b\end{align*}, and \begin{align*}c\end{align*} based upon the order in which the terms were given. Students will assume that \begin{align*}a\end{align*} is the first coefficient, \begin{align*}b\end{align*} the second, and \begin{align*}c\end{align*} the last.

- Remind students rewrite each quadratic equation in standard form first. See Example 3.

General Tip: Have students simplify one step at a time when using the quadratic formula:

- Substitute.
- Simplify under the radical.
- Determine the square root.
- Simplify the numerator.
- Simplify the denominator.
- Divide.

General Tip: Some common mistakes associated with the quadratic formula are:

- Not using the minus sign that goes with a coefficient.

Example: \begin{align*}3x^2 + 7x-4 =0\end{align*}. Use \begin{align*}a = 3, b = 7\end{align*}, and \begin{align*}c = -4. (c \neq 4)\end{align*}

- Losing the minus sign on \begin{align*}-b\end{align*}.

Example: \begin{align*}3x^2 - 7x + 4 =0\end{align*}. Use \begin{align*}a = 3, b = -7\end{align*}, and \begin{align*}c = 4\end{align*}.

\begin{align*}x = \frac{-(-7) \pm \sqrt{(-7)^2 -4(3)(4)}}{2(3)}\end{align*} and not \begin{align*}x = \frac {-7 \pm \sqrt{(-7)^2 -4(3)(4)}}{2(3)}\end{align*}

- Canceling a factor of the denominator with \begin{align*}-b\end{align*} only. Remind students that in order to cancel a factor from the denominator, it must be a common factor in every term in the numerator. Unless that factor comes out of the radical, then the factor in the denominator
*cannot*be canceled.

Example: For \begin{align*}a = 3\end{align*}, \begin{align*} b = 6\end{align*}, and \begin{align*}c = -1\end{align*}, \begin{align*}x = \frac{-6 \pm \sqrt{6^2 -4(3)(-1)}}{2(3)} = \frac{-6 \pm \sqrt{48}} {6} \nRightarrow x = \frac{-\cancel{6} \pm \sqrt{48}}{\cancel{6}}\end{align*}.

- Rounding at steps other than the last step.

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