Solving Quadratic Equations by Completing the Square The Digital SAT Math Guide to Quadratic Equations (Part 2)

 

The  Digital SAT Math Guide to Quadratic Equations (Part 2)

Solving Quadratic Equations by Completing the Square

In the previous chapter, you learned how to solve quadratic equations by factoring. Factoring is often the quickest method, but many quadratic equations on the Digital SAT cannot be factored easily. Some have large coefficients, some produce fractional values, and others have no integer factors at all.

For these equations, completing the square provides a systematic method that always works. Unlike factoring, you do not have to guess factor pairs or recognize patterns. Instead, you follow the same sequence of algebraic steps every time.

Before learning the procedure, remember one important rule.

The coefficient of x² should be 1 before you begin completing the square.

A quadratic equation whose coefficient of x² is 1 is called a monic quadratic equation.

If the equation is not monic, divide every term on both sides of the equation by the coefficient of x². This makes the remaining steps much easier and reduces mistakes.

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Universal Method for Completing the Square

Always follow these steps.

1. Write the equation in standard form.

2. If the coefficient of x² is not 1, divide every term on both sides by that coefficient.

3. Move the constant term to the opposite side.

4. Take half of the coefficient of x.

5. Square that number.

6. Add the squared value to both sides.

7. Rewrite the left side as the square of a binomial.

8. Take the square root of both sides.

9. Remember both the positive and negative square roots.

10. Solve the resulting linear equations.

11. Check every solution in the original equation.

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Example 1

Solve

2x² + 12x + 4 = 0

Step 1

The coefficient of x² is 2.

Divide every term on both sides by 2.

2x² ÷ 2 + 12x ÷ 2 + 4 ÷ 2 = 0 ÷ 2

Simplify.

x² + 6x + 2 = 0

The equation is now monic.

Step 2

Subtract 2 from both sides.

x² + 6x + 2 − 2 = 0 − 2

Simplify.

x² + 6x = −2

Step 3

Take half of 6.

6 ÷ 2 = 3

Square it.

3² = 9

Step 4

Add 9 to both sides.

x² + 6x + 9 = −2 + 9

Simplify.

x² + 6x + 9 = 7

Step 5

Rewrite the left side.

(x + 3)² = 7

Step 6

Take square roots.

√((x + 3)²) = ±√7

Simplify.

x + 3 = ±√7

Step 7

Subtract 3 from both sides.

Positive solution:

x = −3 + √7

Negative solution:

x = −3 − √7

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Example 2

Now solve a question that produces fractions immediately after making the quadratic monic.

4x² + 10x − 3 = 0

Step 1

The coefficient of x² is 4.

Divide every term by 4.

4x² ÷ 4 + 10x ÷ 4 − 3 ÷ 4 = 0 ÷ 4

Simplify.

x² + ⁵⁄₂x − ³⁄₄ = 0

Notice that fractions are perfectly acceptable. Do not convert them to decimals because exact fractions make later calculations more accurate.

Step 2

Move the constant term.

Add ³⁄₄ to both sides.

x² + ⁵⁄₂x = ³⁄₄

Step 3

Take half of the coefficient of x.

The coefficient is ⁵⁄₂.

Half of ⁵⁄₂ is

⁵⁄₂ ÷ 2 = ⁵⁄₄

Now square the result.

(⁵⁄₄)² = ²⁵⁄₁₆

Step 4

Add ²⁵⁄₁₆ to both sides.

x² + ⁵⁄₂x + ²⁵⁄₁₆ = ³⁄₄ + ²⁵⁄₁₆

Convert ³⁄₄ to sixteenths.

³⁄₄ = ¹²⁄₁₆

Now add.

¹²⁄₁₆ + ²⁵⁄₁₆ = ³⁷⁄₁₆

The equation becomes

x² + ⁵⁄₂x + ²⁵⁄₁₆ = ³⁷⁄₁₆

Step 5

Rewrite the left side.

(x + ⁵⁄₄)² = ³⁷⁄₁₆

Step 6

Take square roots.

√((x + ⁵⁄₄)²) = ±√(³⁷⁄₁₆)

Simplify.

x + ⁵⁄₄ = ±√37⁄4

Step 7

Subtract ⁵⁄₄ from both sides.

x = −⁵⁄₄ ± √37⁄4

These are the exact solutions.

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Notice that completing the square works just as well with fractions as it does with whole numbers. On the Digital SAT, leaving answers in exact fractional or radical form is often the correct approach unless the question specifically asks for a decimal approximation.