# 10.4: Quadratic Equations by Completing the Square

## 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:

- Point out that the leading coefficient is .
- In Example 3, . Students learn to factor from the whole expression before completing the square, whether or not the terms are multiples of . 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:*

Solution: Although , and therefore and ; we show that completing the square results in the same solutions.

Rewrite:

Divide by the leading coefficient; this results in an equation with .

Add the constant to both sides of the equation:

Factor the perfect square trinomial and simplify.

Take the square root of both sides:

Answer: and 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 coefficient. Remind students that dividing by is equivalent to multiplying by .

Additional Example:

*Solve the quadratic equation by completing the square:*

Hint: Rewrite the equation:

And divide by the leading coefficient:

To find the constant that must be added to both sides of the equation, *multiply* by and then square:

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

Additional Example:

*Find the vertex of the parabola with equation:*

Solution: Rewrite the equation with minus signs:

The vertex is at , where and .

General Tip: Remind students that the leading coefficient must be 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.