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# 10.5: Quadratic Equations by the Quadratic Formula

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 $x$ 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.

Feb 22, 2012

## Last Modified:

Aug 22, 2014
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CK.MAT.ENG.TE.1.Algebra-I.10.5