# What is the vertex form of y=(x – 12)(x + 8) ?

Jul 24, 2016

Solution method explained in more detail.

color(green)(y=(x-2)^2-100

#### Explanation:

Multiply the brackets giving:

$y = {x}^{2} - 12 x + 8 x - 96 \text{ "->" } y = {x}^{2} - 4 x - 96$.......Equation(1)
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$\textcolor{b l u e}{\text{Step 1}}$

Write as:

$y = \left({x}^{2} - 4 x\right) - 96 + k$
where $k$ is a correction that cancels out the error produced by the following process.
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$\textcolor{b l u e}{\text{Step 2}}$
Take the power from ${x}^{2}$ and move it outside the brackets

$y = {\left(x - 4 x\right)}^{2} - 96 + k$
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$\textcolor{b l u e}{\text{Step 3}}$
Discard the $x$ from $4 x$

$y = {\left(x - 4\right)}^{2} - 96 + k$
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$\textcolor{b l u e}{\text{Step 3}}$
halve the $- 4$ inside the brackets

$y = {\left(x - 2\right)}^{2} - 96 + k \text{ "larr" Now we need the } k$...Equation(2)
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$\textcolor{b l u e}{\text{Step 4}}$

If you were to expand the bracket you would have:
$y = {x}^{2} - 4 x \textcolor{m a \ge n t a}{+ 4} - 96 + k$

The $\textcolor{m a \ge n t a}{+ 4}$ is the error. If you compare this to Equation(1) you will find that the +4 is not in it

So we set $+ 4 + k = 0 \implies k = - 4$

So Equation(2) becomes:

color(brown)(y=(x-2)^2-96+kcolor(blue)(" "->" "y=(x-2)^2-96-4)

color(green)(y=(x-2)^2-100
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$\textcolor{m a \ge n t a}{\text{General case}}$

$y = a {x}^{2} + b x + c \text{ "->" } y = a {\left(x + \frac{b}{2 a}\right)}^{2} + c - \left[a {\left(\frac{b}{2 a}\right)}^{2}\right]$
$\text{ } \uparrow$
$\text{ } k$