Solve the quadratic equation by completing the square

**Sapphire19** · January 25th 2011, 01:47 PM

HERE IS A PROBLEM FROM MYMATHLAB. I WROTE ALL OF THE DIRECTIONS. AT THE END I SAY WHAT I DON'T UNDERSTAND. PLEASE EXPLAIN IT TO ME AS IF YOU ARE EXPLAINING IT TO A FIVE YEAR OLD.

2x^2+ x- 1/8=0. Solve the quadratic equation by completing the square.
1.)We always begin this process by rearranging the equation so that the constant is on the right side.
2x^2+x= 1/8
2.) Next, divide both sides by 2 so that the coefficient of x^2 is 1.
x^2 + 1/2x= 1/16
3.) Now, find the number that completes the square. To find this number, take the square of half of the coefficient of X. What number will complete the square?
1/16. Now add 1/16 to both sides of the equation.

(x + 1/4)^2 = 1/8

4.)Now take the square root of both sides. The right side will need to be simplified.

x+1/4 = √2/4

Now, solve the equation for x.
x= 1/4 +√2/4

I DON'T UNDERSTAND WHERE THE NUMBER √2/4 CAME FROM. HOW DID THAT COME FROM 1/8? IT SAID TO TAKE THE SQUARE ROOT OF BOTH SIDES. HOW IS √2/4 THE ANSWER FOR 1/8?? IT DOESN'T MAKE ANY SENSE. PLEASE HELP ME

**skeeter** · January 25th 2011, 01:55 PM

$\left(x+\frac{1}{4}\right)^2 = \frac{1}{8}<br />$

$x + \frac{1}{4} = \pm \sqrt{\frac{1}{8}} = \pm \frac{1}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \pm \frac{\sqrt{2}}{4}$

$x = -\frac{1}{4} \pm \frac{\sqrt{2}}{4} = \frac{-1 \pm \sqrt{2}}{4}$

**Archie Meade** · January 25th 2011, 01:57 PM

$\displaystyle\sqrt{\frac{1}{8}}=\frac{1}{\sqrt{8}} =\frac{1}{\sqrt{4(2)}}=\frac{1}{\sqrt{4}\sqrt{2}}= \frac{1}{2\sqrt{2}}=\frac{\sqrt{2}}{\sqrt{2}}\;\fr ac{1}{2\sqrt{2}}=\frac{\sqrt{2}}{2(2)}$

Of course, both the positive and negative square roots should be taken.

**Sapphire19** · January 25th 2011, 02:15 PM

can you please explain WHY though? Why does the 1 on top all of a sudden turn into a 2 after 1/2√2

**Archie Meade** · January 25th 2011, 02:33 PM

Originally Posted by Sapphire19

can you please explain WHY though? Why does the 1 on top all of a sudden turn into a 2 after 1/2√2

It's just another way to write the same value.

When we had $\frac{1}{2\sqrt{2}}$

convention is to write it in the form $\frac{a}{b}$

with a surd on top, instead of under the line.

We "resolve" it using $\sqrt{2}\sqrt{2}=2$

However, we can only multiply the fraction by 1, so it is multiplied by $\frac{\sqrt{2}}{\sqrt{2}}$

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