   SEARCH HOME Math Central Quandaries & Queries  Question from rebecca, a student: Find the roots of the equation by Completing the square. Express your answer in exact form and in decimal form to 2 decimal places. a) x(squared)+10x+23 b)3y(squared)+12y+3 Hi Rebecca.

I'll show you an example and then you can use this method to solve your questions.

Example: Find the roots of 2x² + 16x + 20 by completing the square.

Completing the square means we want to make a perfect square as we do this.

We'll solve for it using the equation

2x² + 16x + 20 = 0

First things first though: simplify the numbers if you can! 2 goes into everything, so it reduces to just:

x² + 8x + 10 = 0

I start by moving the constant (third) term to the other side:

x² + 8x = -10

Now I want the left hand side to become a perfect square. The way I do that is by taking half the co-efficient of the x term (that is, one half of 8, which is 4) and then squaring that (to make 16). I add this to both sides of the equation:

x² + 8x + 16 = -10 + 16
x² + 8x + 16 = 6

The left hand side is now a perfect square. It's the perfect square of x and half of +8 (which is 4). So I can re-write this as:

(x + 4)² = 6

Now I take the square root of both sides. That just cancels out the squared on the left side. Remember that there is both a positive and negative square root!

x + 4 = ±√6

I solve for x:

x = ±√6 - 4

So this is actually two numbers (because of that ± sign). The two roots are:

x = √6 - 4
x = - √6 - 4

Now my example is done. I hope you can see how to do your problems now, step by step.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.