Lesson 12: Completing the Square (Part 1)

Completing the Square (Part 1)

Student Summary

Turning an expression into a perfect square can be a good way to solve a quadratic equation. Suppose we wanted to solve $x^2 - 14x +10 = \text-30$ .

The expression on the left, $x^2 - 14x +10$ , is not a perfect square, but $x^2 - 14x + 49$ is a perfect square. Let’s transform that side of the equation into a perfect square (while keeping the equality of the two sides).

One helpful way to start is by first moving the constant that is not a perfect square out of the way. Let’s subtract 10 from each side:

$\displaystyle \begin{aligned} x^2 - 14x +10 - 10 &= \text-30 - 10\\ x^2 - 14x &= \text-40 \end{aligned}$

And then add 49 to each side:

$\displaystyle \begin{aligned} x^2 - 14x +49 &= \text-40 +49\\ x^2 - 14x+49 &= 9 \end{aligned}$

The left side is now a perfect square because it’s equivalent to $(x-7)(x-7)$ or $(x-7)^2$ . Let’s rewrite it:

$\displaystyle (x-7)^2=9$

If a number squared is 9, the number has to be 3 or -3. Solve to finish up:

$\displaystyle \begin{aligned} x-7=3 \quad & \text{or} \quad x-7=\text-3\\ x=10 \quad & \text{or} \quad x=4 \end{aligned}$

This method of solving quadratic equations is called completing the square. In general, perfect squares in standard form look like $x^2 + bx + \left(\frac{b}{2} \right)^2$ , so to complete the square, take half of the coefficient of the linear term and square it.

In the example, half of -14 is -7, and $(\text-7)^2$ is 49. We wanted to make the left side $x^2 - 14x + 49.$ To keep the equation true and maintain equality of the two sides of the equation, we added 49 to each side.