# 10.5: Quadratic Equations by the Quadratic Formula

## 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 squared can be found by
*multiplying*by :

- 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 , , and based upon the order in which the terms were given. Students will assume that is the first coefficient, the second, and 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: . Use , and

- Losing the minus sign on .

Example: . Use , and .

and not

- Canceling a factor of the denominator with 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 , , and , .

- Rounding at steps other than the last step.