# Completing The Square

## Completing the square definition

Algebra and geometry are closely connected. Geometry, as in coordinate graphing and polygons, can help you make sense of algebra, as in quadratic equations.Â **Completing the square**Â is one additional mathematical tool you can use for many challenges:

Simplify algebraic expressions

Solve quadratic equations

Convert expressions from one form to another

Find the minimum or maximum values of quadratic functions

When completing the square, we can take a quadratic equation like this, and turn it into this:

## Completing the square

"**Completing the square**" comes from the exponent for one of the values, as in this simpleÂ **binomial expression**:

We useÂ $b$Â for the second term because we reserveÂ aaÂ for the first one. We might have hadÂ $a{x}^{2}$, but if $a$Â is **1**, you have no need to write it.

Anyway, you have no idea what valuesÂ $x$Â orÂ $b$Â have, so how can you proceed? You already knowÂ $x$Â will be multiplied times itself, to begin.

Think about a square in geometry. You have four congruent-length sides, with an enclosed area that comes from multiplying a number times itself. In this expression,Â $x$Â timesÂ $x$Â is a square with an area ofÂ ${x}^{2}$:

Hold onÂ â€“ we still have unknown variableÂ $b$Â timesÂ $x$. What would that look like? That would be a rectangleÂ $x$Â units tall andÂ $b$Â units wide, attached to ourÂ ${x}^{2}$ square:

To make better sense of that rectangle, divide it equally between the width and length of theÂ ${x}^{2}$Â square. That would make each rectangleÂ $\frac{b}{2}$Â timesÂ $x$:

That means the new almost-square is $x+\frac{b}{2}$, but we are missing a tiny corner, which would have a value of $\frac{b}{2}$ times itself, or ${\left(\frac{b}{2}\right)}^{2}$:

That last step literally completed the square, so now we have this:

This refines or simplifies to:

You need to also subtractÂ ${\left(\frac{b}{2}\right)}^{2}$Â if you are, in fact, trying to work an equation (you cannot add something without balancing it by subtracting it). In our case, we were just showing how the square is really a square, in a geometric sense.

### Completing the square formula

Here is a more complete version of the same thing:

As soon as you seeÂ * x*Â raised to a power, you know you are dealing with a candidate for "completing the square."

The role ofÂ bbÂ from our earlier example is played here by theÂ **2**. We added a value,Â **+3**, so now we have aÂ **trinomial expression**.

${x}^{2}+2x+3$ is rewritten as:

So, divideÂ * b*Â byÂ

*Â and square it, which you then add and subtract to get:*

**2**Now, you can simplify as:

Which is equal to:

This simplifies to:

On a graph, this plots a parabola with a vertex atÂ **(-1,Â 2)**.

## How to complete the square

You can use completing the square toÂ **simplify algebraic expressions**. Here is a straightforward example with steps:

Divide the middle term,Â **20*** x*, byÂ

**2**Â and square it, then both add and subtract it:

Simplify the expression:

### Steps to completing the square

Seven steps are all you need to complete the square in anyÂ **quadratic equation**. The general form of a quadratic equation looks like this:

Completing the square steps:

Isolate the number or variable

**c**Divide all terms by

(the coefficient of ${x}^{2}$, unless ${x}^{2}$ has no coefficient).**a**Divide coefficient

by two and then square it.**b**Add this value to both sides of the equation.

Rewrite the left side of the equation in the form ${\left(x+d\right)}^{2}$ where $d$ is the value of $\frac{b}{2}$ you found earlier.

Take the square root of both sides of the equation; on the left side, this leaves you with $x+d$.

Subtract whatever number remains on the left side of the equation to yield

**x****complete the square**.

## Completing the square examples

We will provide three examples of quadratic equations progressing from easier to harder. Give each a try, following the seven steps described above. The first one does not place a coefficient with ${x}^{2}$.

${x}^{2}+3x-4=0$

${x}^{2}+3x=4$

${x}^{2}+3x+{\left(\frac{3}{2}\right)}^{2}=4+{\left(\frac{3}{2}\right)}^{2}$

${\left(x+\frac{3}{2}\right)}^{2}=\frac{25}{4}$

$x+\frac{3}{2}=-\sqrt{\frac{25}{4}}$

$x+\frac{3}{2}=\sqrt{\frac{25}{4}}$

$x=1$ and $x=-4$.

### Solving quadratic equations by completing the square

Our second example uses a coefficient withÂ ${x}^{2}$ for solving a quadratic equation by completing the square:

$2{x}^{2}-4x-2=0$

$2{x}^{2}-4x=2$

${x}^{2}-2x=1$

${x}^{2}-2x+{\left(\frac{-2}{2}\right)}^{2}=1+{\left(\frac{-2}{2}\right)}^{2}$

${x}^{2}-2x+{\left(-1\right)}^{2}=1+{\left(-1\right)}^{2}$

${x}^{2}-2x+{\left(-1\right)}^{2}=2$

${\left(x-1\right)}^{2}=2$

$x-1=-\sqrt{2}$

$x-1=\sqrt{2}$

$x=-\sqrt{2+1}$ and $x=\sqrt{2+1}$.

### Challenge example

Our third example is all bells and whistles with really big numbers. See how you do!

$20{x}^{2}-30x-40=0$

$20{x}^{2}-30x=40$

${x}^{2}-1.5x=2$

${x}^{2}-1.5x+{\left(\frac{-1.5}{2}\right)}^{2}=2+{\left(\frac{-1.5}{2}\right)}^{2}$

${x}^{2}-1.5x+{\left(0.75\right)}^{2}=2+{\left(0.75\right)}^{2}$

${x}^{2}-1.5x+{\left(-0.75\right)}^{2}=\frac{41}{16}$

${\left(x-0.75\right)}^{2}=\frac{41}{16}$

$x-0.75=-\sqrt{\frac{41}{16}}$

$x-0.75=\sqrt{\frac{41}{16}}$

$x=\frac{\left(-\sqrt{41+3}\right)}{4}$ and $x=\frac{\left(\sqrt{41+3}\right)}{4}$.