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.
Terms introduced in this lesson:
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.
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:
- Simplify under the radical.
- Determine the square root.
- Simplify the numerator.
- Simplify the denominator.
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 .
- 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.