# Completing the square in a quadratic equation

## Completing the square in a quadratic equation

The process of completing the square is a way to solve a quadratic equation. This procedure converts an equation written in the standard form of the quadratic function $ax^2+bx+c$ into an expression with a variable squared, as in the following example: $(X-r)^2-w$ where $r$ and $w$ are parameters.

### Steps of the completing the square procedure -> combined in an example

Given the function $X^2+10x+9$

1. Let's observe the quadratic function and focus only and exclusively on $ax^2+bx$.
For now, we will ignore $C$.
In the example, we will focus on $X^2+10x$

2. Let's remember the formulas for shortcut multiplication and ask ourselves what expression we could place inside the parentheses squared, that is, what $(a-b)^2$  or $(a+b)^2$ as appropriate, that gives us what appears in the pair we are focusing on $ax^2+bx$

In the example
the convenient shortcut multiplication formula is: $a^2+2ab+b^2=(a+b)^2$

Let's see what we can replace $a$ and $b$ with to obtain $X^2+10x$?
The answer is $(X+5)^2$
we will open this expression according to the shortcut multiplication formula and obtain: $X^2+10x+25$

3. Notice that, the expression inside the parentheses also brings with it some number and not only the pair we are focusing on, therefore, we must neutralize it. If the added number is negative, we will add it to the equation to cancel it out. If the number is positive, we will subtract it from the equation and, in this way, it will be canceled out.

Also, we will return to $C$ in the original function and also write it in the equation.
In the example:
$X^2+8x+25$

the number $25$ has been added. To cancel it out we will subtract $25$ (without adding) and we will not forget about $C$   from the original equation $9$.

We will obtain:
$X^2+8x-25+9=$
4. Let's replace the pair $ax^2+bx$ with the corresponding expression in parentheses squared that we have found and order the equation: we will complete the square.

In the example:
$(X+5)^2-25+9=$
$(X+5)^2-16$

Now :
The steps to solve the quadratic equation after completing the square: let's set the equation to zero.

In the example:
$(X+5)^2-16=0$

Let's move the independent variable to the second term.

In the example :

$(X+5)^2=16$

We will write the independent variable as a number squared.
In the example:
$(X+5)^2=4^2$
Let's solve the equation and see how many possible solutions there are.

In the example :
$(X+5)^2=4^2$

We will see that we have $2$ solutions and solve:
Solution number one:
$X+5=4$
$X=-1$

Solution two:
$X+5=-4$
$X=-9$

The results are:
$X=-1,-9$

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