Completing the square - example 2

Complete the square $2x^2-6x+3=0$

Quick sketch

NOTE:

The coefficient (or number) in front of the $x^2$ must be a one.

So divide both sides by 2.

$(2x^2)/2-(6x)/2+3/2=0/2$

$x^2-3x+1\1/2=0$

Remember

Complete the square and split the square into quarters using x square

Fill the square in with the next term

Is the same as

This table is the same as the one above

Fill in the table

Fill in the table multiplying the y axis by the x axis

Is the same as

This table will mean the same thing as the one above

NOTE:

This is the same as $(x-3/2)^2$

If you add up each area you get:

$x^2-3/2x-3/2x+9/4$

$x^2-6/2x+9/4$

$x^2-3x+9/4$

$x^2-3x+2\1/4$

Originally we had $x^2-3x+1\1/2=0$

Now we have $x^2-3x+2\1/4=0$

Always plot this on a number line.

Use the number line to work out the difference between the original number and the new number

The number line will help you remember

Original number $-$ New number

$1\1/2-2\1/4=-3/4$ (We need to $-3/4$ )

So $x^2-3x+1\1/2=0$

Is the same as $(x-3/2)^2-3/4=0$

Which can now be solved

$(x-3/2)^2-3/4=0$

$(x-3/2)^2=3/4$

$(x-3/2)=+-sqrt(3/4)$

(Don't forget the root of anything can be $+$ or $-$ )

$x-3/2=+-sqrt(3/4)$

$x=3/2+-sqrt(3/4)$

$x=3/2+sqrt(3/4)$ or $x=3/2-sqrt(3/4)$

Using calculator

$x=1.5+0.866$ or $1.5-0.866$

$x=2.366$ or $0.634$

Now check

$2x^2-6x+3=0$

If $x=2.366$ $2times2.366^2-6times2.366+3=0$

$11.19-14.19+3=0$ Which is correct

If $x=0.634$ $2times0.634^2-6times0.634+3=0$

$0.804-3.804+3=0$ Which is correct

Answer:

The roots of $2x^2-6x+3=0$ are $x=2.366$ or $0.634$