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# 10.4: Quadratic Equations by Completing the Square

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## Learning Objectives

At the end of this lesson, students will be able to:

• Complete the square of a quadratic expression.
• Solve quadratic equations by completing the square.
• Solve quadratic equations in standard form.
• Graph quadratic equations in vertex form.
• Solve real-world problems using functions by completing the square.

## Vocabulary

Terms introduced in this lesson:

completing the square
perfect square trinomial
quadratic equations in standard form
quadratic equations in vertex form
parabola turns up, turns down

## Teaching Strategies and Tips

Use Example 1 to introduce completing the square.

• Remind students of binomial expansions to help them understand how a constant term turns the expression into a perfect square trinomial:

$(x \pm a)^2 = x^2 \pm 2ax+a^2$

• Point out that the leading coefficient is $1$.
• In Example 3, $a \neq 1$. Students learn to factor $a$ from the whole expression before completing the square, whether or not the terms are multiples of $a$. Complete the square of the resulting expression in parentheses.

Give completing the square geometrical meaning.

• Use squares and rectangles for each term of the expression.
• See paragraph preceding Example 4.
• Have students make up a quadratic expression and ask them to complete the square in the geometrical interpretation as an assignment.

There are several reasons to have students learn completing the square:

• It is used to derive the quadratic formula.
• Quadratic equations can be rewritten in vertex form.
• Equations of circles can be rewritten in graphing form.
• Necessary in calculus.

Emphasize that completing the square finds roots

• Regardless of whether the roots are integers, rational or irrational numbers.
• Without having to list all the cases, unlike factoring.

Example:

Solve the following quadratic equation:

$2x^2+x-3=0$

Solution: Although $2x^2+x-3=(2x+3)(x-1)$, and therefore $x= - \frac{3}{2}$ and $x=1$; we show that completing the square results in the same solutions.

Rewrite:

$2x^2+x=3$

Divide by the leading coefficient; this results in an equation with $a = 1$.

$x^2 + \left (\frac{1}{2} \right )x=\frac{3}{2}$

Add the constant to both sides of the equation:

$x^2 + \left (\frac{1}{2} \right )x+ \left (\frac{1}{4} \right )^2 =\frac{3}{2} = \left (\frac{1}{4} \right )^2$

Factor the perfect square trinomial and simplify.

$\left (x + \frac{1}{4} \right )^2 & = \frac{3}{2} + \frac{1}{16} \\\left (x + \frac{1}{4} \right )^2 & = \frac{24}{16} + \frac{1}{16} \\\left (x + \frac{1}{4} \right )^2 & = \frac{25}{16}$

Take the square root of both sides:

$\sqrt{\left (x + \frac{1}{4} \right )^2} & =\sqrt{\frac{25}{16}} \\x + \frac{1}{4} & = \pm \frac{5}{4} \\ x & = - \frac{1}{4} \pm \frac{5}{4} \\ x & = \frac{-1 \pm 5}{4} \\\Rightarrow x & = \frac{-1 + 5}{4} = 1 \\ \Rightarrow x & = \frac{-1 - 5}{4} = - \frac{3}{2}$

Answer: $x =- \frac{3}{2}$ and $x=1$ as expected.

Use Examples 7-9 to show how completing the square helps in graphing quadratic functions.

## Error Troubleshooting

Dividing by the leading coefficient in Review Questions 7, 8, and 14 results in a fractional $b$ coefficient. Remind students that dividing $b$ by $2$ is equivalent to multiplying $b$ by $1/2$.

Solve the quadratic equation by completing the square:

$3x^2 + 5x-3=0$

Hint: Rewrite the equation:

$3x^2 + 5x=3 .$

And divide by the leading coefficient:

$x^2 + \frac{5}{3} x=1$

To find the constant that must be added to both sides of the equation, multiply $5/3$ by $1/2$ and then square:

$\left (\frac{5}{3} \cdot \frac{1}{2} \right )^2 = \left (\frac{5}{6} \right )^2$

General Tip: Remind students to rewrite the vertex form of a quadratic equation with minus signs to find $h$ and $k$ correctly. See Examples 7-9.

Find the vertex of the parabola with equation:

$y+3= -2(x-5)^2$

Solution: Rewrite the equation with minus signs:

$y-(-3) = -2(x-5)^2$

The vertex is at $(h,k)$, where $h = -3$ and $k = 5$.

General Tip: Remind students that the leading coefficient must be $1$ before completing the square. They will need to divide or factor to accomplish this.

General Tip: Suggest that students rewrite equations in standard form before completing the square.

In Examples 10 and 11 and Review Questions 25 and 26, have students round in the last step.

## Date Created:

Feb 22, 2012

Aug 22, 2014
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