Completing the Square: Complete Method

Article Contents

1. What is Completing the Square
2. Basic Procedure
3. Deriving the Formula
4. Examples
5. Quadratic Equations
6. Interactive Exercises

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1. What is Completing the Square {#what-is-completing}

Completing the square is a technique that converts a quadratic expression to the form $(x+p{)}^2+q$.

For example:

$$x^2+6x+5=(x+3{)}^2-4$$

Why is it Useful?

• Solving quadratic equations
• Finding the vertex of a parabola
• Integration (later in mathematics)

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2. Basic Procedure {#basic-procedure}

For an expression $x^2+bx$:

1. The coefficient of $x$ is $b$
2. Half of $b$ is $\frac{b}{2}$
3. Add and subtract $(\frac{b}{2}{)}^2$
$$x^2+bx=x^2+bx+(\frac{b}{2}{)}^2-(\frac{b}{2}{)}^2=(x+\frac{b}{2}{)}^2-(\frac{b}{2}{)}^2$$

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3. Deriving the Formula {#deriving}

Example $x^2+6x$

Coefficient $b=6$

$$\frac{b}{2}=3,(\frac{b}{2}{)}^2=9$$
$$x^2+6x+9-9=(x+3{)}^2-9$$

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4. Examples {#examples}

Example 1: $x^2+4x+1$

$$x^2+4x+1=(x^2+4x+4)-4+1=(x+2{)}^2-3$$

Example 2: $x^2-8x+5$

$$x^2-8x+5=(x^2-8x+16)-16+5=(x-4{)}^2-11$$

Example 3: $2x^2+8x+3$

First factor out 2:

$$2(x^2+4x)+3=2[(x^2+4x+4)-4]+3=2(x+2{)}^2-8+3=2(x+2{)}^2-5$$

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5. Quadratic Equations {#quadratic-equations}

Solving $x^2+6x+5=0$

By completing the square:

$$x^2+6x+9=-5+9$$
$$(x+3{)}^2=4$$
$$x+3=\pm 2$$
$$x=-1\text{or}x=-5$$

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6. Interactive Exercises {#interactive-exercises}

Practice completing the square:

• Complete the Square
• Difference of Squares
• Expanding

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Summary

Expression	Completed Form
$x^2+bx$	$(x+\frac{b}{2}{)}^2-(\frac{b}{2}{)}^2$
$x^2+6x+5$	$(x+3{)}^2-4$
$x^2-8x+5$	$(x-4{)}^2-11$

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Related Articles

• Difference of Squares – related technique
• Introduction to Algebra – basics
• Perfect Cube – advanced formulas