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10.5: Quadratic Equations by the Quadratic Formula

Difficulty Level: 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.


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 x squared can be found by multiplying by 1/2:


  • 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 a, b, and c based upon the order in which the terms were given. Students will assume that a is the first coefficient, b the second, and c 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: 3x2+7x4=0. Use a=3,b=7, and c=4.(c4)

  • Losing the minus sign on b.

Example: 3x27x+4=0. Use a=3,b=7, and c=4.

x=(7)±(7)24(3)(4)2(3) and not x=7±(7)24(3)(4)2(3)

  • Canceling a factor of the denominator with b 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 a=3, b=6, and c=1, x=6±624(3)(1)2(3)=6±486x=6±486.

  • Rounding at steps other than the last step.

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