# How do you solve x^2 - 4x - 9 = 0 by completing the square?

Jun 6, 2017

$x = 2 \pm \sqrt{13}$

#### Explanation:

$\text{to complete the square}$

add (1/2"coefficient of x-term")^2 " to both sides"

$\text{that is add " (-4/2)^2=4" to both sides}$

$\Rightarrow {x}^{2} - 4 x \textcolor{red}{+ 4} - 9 = 0 \textcolor{red}{+ 4}$

$\Rightarrow {\left(x - 2\right)}^{2} - 9 = 4$

$\Rightarrow {\left(x - 2\right)}^{2} = 13$

$\textcolor{b l u e}{\text{take the square root of both sides}}$

$\sqrt{{\left(x - 2\right)}^{2}} = \pm \sqrt{13} \leftarrow \text{ note plus or minus}$

$\Rightarrow x - 2 = \pm \sqrt{13}$

$\Rightarrow x = 2 \pm \sqrt{13}$

$\text{or " x~~5.6, x~~-1.6" to 1 decimal place}$

Jun 6, 2017

$x = + 5.606 \mathmr{and} x = - 1.606$

#### Explanation:

${x}^{2} - 4 x - 9 = 0$

To complete the square means exactly what it says...

"make an expression into a perfect square by adding the part that is missing..."

The steps are given, refer to the details of the working below:

In $a {x}^{2} + b x + c = 0$

$1 : \rightarrow \text{ "1x^2-4x" } \textcolor{red}{- 9} = 0$

$2 : \rightarrow \text{ "x^2 color(blue)(-4)x" } = \textcolor{red}{9}$

$3 : \rightarrow \text{ "x^2 -4x " } \textcolor{b l u e}{+ 4} = 9 \textcolor{b l u e}{+ 4}$

$4 : \rightarrow \text{ "(x-2)^2" } = 13$

$5 : \rightarrow \text{ "x-2" } = \pm \sqrt{13}$

$6 : \rightarrow \text{ } x \textcolor{w h i t e}{w w w w w w w} = + \sqrt{13} + 2 = + 5.606$
$6 : \rightarrow \text{ } x \textcolor{w h i t e}{w w w w w w w} = - \sqrt{13} + 2 = - 1.606$

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• Step 1: Make a=1" " (a is already equal to $1$ )

• Step 2: Move the constant to the other side.

• Step 3: complete the square by adding $\textcolor{b l u e}{{\left(\frac{b}{2}\right)}^{2}}$ to both sides.
In this case $\textcolor{b l u e}{b = - 4} \text{ }$ so, $\textcolor{b l u e}{{\left(\frac{- 4}{2}\right)}^{2} = {\left(- 2\right)}^{2} = + 4}$

• Step 4: Write the LHS as the square of a binomial

• Step 5: square root both sides $\rightarrow$ remember +-sqrt
• Step 6: solve for $x$ to get 2 values