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10.5: Quadratic Equations by the Quadratic Formula

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

\left (\frac{b}{a} \cdot \frac{1}{2} \right )^2 = \left (\frac{b}{2a} \right )^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: 3x^2 + 7x-4 =0. Use a = 3, b = 7, and c = -4. (c \neq 4)

  • Losing the minus sign on -b.

Example: 3x^2 - 7x + 4 =0. Use a = 3, b = -7, and c = 4.

x = \frac{-(-7) \pm \sqrt{(-7)^2 -4(3)(4)}}{2(3)} and not x = \frac {-7 \pm \sqrt{(-7)^2 -4(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 = \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}}.

  • Rounding at steps other than the last step.

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