Math CentralQuandaries & 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

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.

Stephen La Rocque.

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