Completing the Square

Key Questions

How do I complete the square if the coefficient of $x^2$ is not 1?

Remember that

$(ax + b)^2 = a^2x^2 + 2abx + b^2$

So the general idea to get $b^2$ is by getting $x$ 's coefficient,
dividing by $2a$ , and squaring the result.

Example

$3x^2 + 12x$

$a = 3$
$2ab = 12$
$6b = 12$
$b = 2$
$3x^2 + 12x + 4 = (3x + 2)^2$

You can also factor out $x^2$ 's coefficient.. and proceed with completing the square.

Example:
$2X^2 +4X$

$2(X^2 + 2X)$

$2(X + 1)^2$

seph · · Oct 7 2014
How do I complete the square?

Answer:

$ax^2+bx+c = a(x+b/(2a))^2 + (c - b^2/(4a))$

The secret is that $b/(2a)$ bit

Explanation:

Suppose you are given a quadratic equation to solve:

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

..which is in the form..

$ax^2+bx+c = 0$ with $a = 2$ , $b=-3$ and $c=-2$

$b/(2a) = -3/4$

So we find:

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

$=2(x^2-(3x)/2+9/16)$

$=2x^2-3x+9/8$

So:

$2(x-3/4)^2-25/8 = 2(x-3/4)^2-9/8-2$

$=2x^2-3x+9/8-9/8-2$

$=2x^2-3x-2$

So:

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

turns into:

$2(x-3/4)^2-25/8 = 0$

Hence:

$(x-3/4)^2 = 25/16$

So:

$x-3/4 = +-sqrt(25/16) = +-5/4$

and

$x = 3/4+-5/4$

George C. · 2 · Oct 23 2015
Does completing the square always work?

Answer:

In order to use the Completing the Square method, the value for $a$ in the quadratic equation must be $1$ . If it is not $1$ , you will have to use the AC method or the quadratic formula in order to solve for $x$ .

Explanation:

Completing the square is a method used to solve a quadratic equation, $ax^2+bx+c$ , where $a$ must be $1$ . The goal is to force a perfect square trinomial on one side and then solving for $x$ by taking the square root of both sides.

The method is explained at the following website:

http://www.regentsprep.org/regents/math/algtrig/ate12/completesqlesson.htm

Meave60 · 5 · Nov 30 2015
What does completing the square mean?

Completing the square is a method that represents a quadratic equation as a combination of quadrilateral used to form a square.

The basis of this method is to discover a special value that when added to both sides of the quadratic that will create a perfect square trinomial.

That special value is found by evaluation the expression $(b/2)^2$ where $b$ is found in $ax^2+bx+c=0$ . Also in this explanation I assume that $a$ has a value of $1$ .

$ax^2+bx+(b/2)^2=(b/2)^2-c$

$(ax+b/2)^2=(b/2)^2-c$

$sqrt((ax+b/2)^2)=+-sqrt((b/2)^2-c)$

$ax+b/2=+-sqrt((b/2)^2-c)$

$ax=+-sqrt((b/2)^2-c)-b/2$

A quadratic that is a perfect square is very easy to solve.

Please take a look at the video to see an example of completing the square visually.

Completing the Square

AJ Speller · 1 · Sep 16 2014