# What is the vertex form of y= - x^2 - 10x + 20 ?

Dec 12, 2015

$y = - {\left(x + 5\right)}^{2} + 45$

#### Explanation:

Vertex form of a parabola: $y = a {\left(x - h\right)}^{2} + k$

In order to put a parabola into vertex form, use the complete the square method.

$y = - {x}^{2} - 10 x + 20$

y=-(x^2+10x+?)+20

Add the value that will cause the portion in parentheses to be a perfect square.

y=-(x^2+10x+25)+20+?

Since we added $25$ inside the parentheses, we must balance the equation.

Notice that the $25$ is ACTUALLY $- 25$ because of the negative sign in front of the parentheses. To balance the $- 25$, add $25$ to the same side of the equation.

$y = - {\left(x + 5\right)}^{2} + 45$

This is the equation in standard form. It also tells you that the vertex of the parabola is $\left(h , k\right)$, or $\left(- 5 , 45\right)$.

Dec 12, 2015

$y = {\left(- x \textcolor{g r e e n}{- 5}\right)}^{2} + \textcolor{b r o w n}{45}$

#### Explanation:

By using the vertex form (completing the square) you introduce an error. If this error is '+some value' then you correct by including '- the same value'

Given: $\textcolor{b l u e}{y = - {x}^{2} - 10 x + 20} \ldots \ldots \ldots \ldots \left(1\right)$

Consider just the right hand side

write as $- 1 \times \textcolor{b l u e}{\left({x}^{2} + 10 x\right)} + 20. \ldots \ldots . . \left(2\right)$
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Now consider just the brackets part}}$

Write instead : ${\left(x + \frac{10}{2}\right)}^{2} \to {\left(x + 5\right)}^{2}$

Multiplying ${\left(x + 5\right)}^{2}$ out and you get:

$\textcolor{b l u e}{\textcolor{red}{\left({x}^{2} + 10 x + 25\right)} < - - - \text{Introduced an error of } 25}$

Using this to replace the brackets in expression (2)

color(blue)(-1xxcolor(red)((x^2+10x+25))+20))

We have gained the extra value of $\textcolor{b l u e}{- 1 \times} \textcolor{red}{25} = - 25$

so it is $\underline{\textcolor{red}{\text{NOT CORRECT}}}$ to write $y = - {\left(x + 5\right)}^{2} + 20$

However, it $\underline{\textcolor{g r e e n}{\text{IS CORRECT}}}$ to write $y = - {\left(x + 5\right)}^{2} \textcolor{g r e e n}{+ 25} + 20$

Giving the final answer of $\textcolor{w h i t e}{. .} y = - {\left(x + 5\right)}^{2} + 45$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$y = {\left(- x \textcolor{g r e e n}{- 5}\right)}^{2} + \textcolor{b r o w n}{45}$

$\textcolor{g r e e n}{\text{Notice that "x_("vertex") =-5" as in the brackets}}$

$\textcolor{b r o w n}{\text{and that "y_("vertex")=45" as the final constant}}$ 