Factoring and Completing the Square

What’s a Quadratic?

A polynomial of degree 2:

$ax^2 + bx + c$

We want to find where it equals zero.

Method 1: Factoring

Find two numbers that multiply to $c$ and add to $b$.

Example: Solve $x^2 + 5x + 6 = 0$

Numbers that multiply to 6 and add to 5: 2 and 3

So $x = -2$ or $x = -3$.

Factoring is fast, but only works when the quadratic factors nicely.

What about $x^2 + 6x + 5 = 0$? Or $x^2 - 4x - 1 = 0$?

We need a method that always works.

Method 2: Completing the Square

Turn the left side into a perfect square, then solve.

The Key Insight

Recall the perfect square pattern:

$(x + k)^2 = x^2 + 2kx + k^2$

If you have $x^2 + bx$, you can complete it by adding $\left(\dfrac{b}{2}\right)^2$.

$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$

The Process

1. Move the constant to the right side
2. Add $\left(\dfrac{b}{2}\right)^2$ to both sides
3. Factor the left side as a perfect square
4. Take the square root of both sides
5. Solve for $x$

Example: Solve $x^2 + 6x + 5 = 0$

Step 1: Move the constant.

$x^2 + 6x = -5$

Step 2: Half of 6 is 3. Square it: 9. Add to both sides.

$x^2 + 6x + 9 = -5 + 9$

$x^2 + 6x + 9 = 4$

Step 3: Left side is a perfect square.

$(x + 3)^2 = 4$

Step 4: Take square root.

$x + 3 = \pm 2$

Step 5: Solve.

$x = -3 + 2 = -1 \quad \text{or} \quad x = -3 - 2 = -5$

Example: Solve $x^2 - 4x - 1 = 0$

Step 1: Move the constant.

$x^2 - 4x = 1$

Step 2: Half of $-4$ is $-2$. Square it: 4. Add to both sides.

$x^2 - 4x + 4 = 1 + 4$

$(x - 2)^2 = 5$

Step 3: Take square root.

$x - 2 = \pm\sqrt{5}$

Step 4: Solve.

$x = 2 + \sqrt{5} \quad \text{or} \quad x = 2 - \sqrt{5}$

When $a \neq 1$

If the coefficient of $x^2$ isn’t 1, divide everything by it first.

Example: Solve $2x^2 + 8x + 6 = 0$

Divide by 2:

$x^2 + 4x + 3 = 0$

Now complete the square:

So $x = -1$ or $x = -3$.

Factoring vs Completing the Square