completing the square

**princess_anna57** · July 11th 2008, 06:45 PM

Solve by completing the square:

x^2 - 3x + 2 = 0

Please show step-by-step instructions because this is the first time I've come across this concept and I've looked through some examples but I can't determine the number that turns the left hand side of the equation into a perfect square?

**topsquark** · July 11th 2008, 06:56 PM

Originally Posted by princess_anna57

Solve by completing the square:

x^2 - 3x + 2 = 0

Please show step-by-step instructions because this is the first time I've come across this concept and I've looked through some examples but I can't determine the number that turns the left hand side of the equation into a perfect square?

You know that
$x^2 + 2ax + a^2 = (x + a)^2$
right?

So we want to find a way to put your equation into that form. The first step is to move the constant term to the other side of the equation:
$x^2 - 3x = -2$

Now, we want to add something to the left hand side to make it a perfect square. Look at that $x^2 + 2ax + a^2$ form again. If we can find what a is, then we need to simply add $a^2$ . So look at the linear term, -3x in your case. We want to find a value for a, so set
$-3x = 2ax$

$a = -\frac{3}{2}$

So we need to add
$a^2 = \left ( -\frac{3}{2} \right ) ^2 = \frac{9}{4}$
to the left hand side. But what we do to one side of the equation, we need to do to the other. Thus
$x^2 - 3x + \frac{9}{4} = -2 + \frac{9}{4}$

The left hand side is now in perfect square form. Simplifying the left and right hand sides gives:
$\left ( x - \frac{3}{2} \right ) ^2 = \frac{1}{4}$

Can you solve it from here?

-Dan

**princess_anna57** · July 11th 2008, 07:17 PM

Is it:

x - 3/2 = + square root 1 / 4 and x - 3/2 = - square root 1/4

then

x = 3/2 + square root 1 /4 and x = 3/2 - square root 1/4

so x = 1.75 or x = 1.25

**kalagota** · July 11th 2008, 07:31 PM

Originally Posted by princess_anna57

Is it:

x - 3/2 = + square root 1 / 4 and x - 3/2 = - square root 1/4

then

x = 3/2 + square root 1 /4 and x = 3/2 - square root 1/4

so x = 1.75 or x = 1.25

no.. what is $\sqrt{\frac{1}{4}}$ ?

**masters** · July 11th 2008, 07:57 PM

Originally Posted by princess_anna57

Is it:

x - 3/2 = + square root 1 / 4 and x - 3/2 = - square root 1/4

then

x = 3/2 + square root 1 /4 and x = 3/2 - square root 1/4

so x = 1.75 or x = 1.25

$x=\frac{3}{2}+\frac{1}{2}=2$

$x=\frac{3}{2}-\frac{1}{2}= 1$

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