# How do you write y=-x^2+20x-80 in vertex form?

Apr 26, 2017

$y = - {\left(x - 10\right)}^{2} + 20 \text{ }$ is the vetex form.

#### Explanation:

The vertex form of a quadratic equation is :
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$\textcolor{b l u e}{y = a {\left(x - h\right)}^{2} + k} \text{ }$ where $\left(h , k\right) \text{ }$is the vertex
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The quadratic form is performed by factorization.
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$y = - {x}^{2} + 20 x - 80$
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$\Rightarrow y = - \left({x}^{2} - 20 x + 80\right)$
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$\Rightarrow y = - \left({\left(x\right)}^{2} - 2 \left(x\right) \left(10\right) + 80\right)$
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To complete the square of the above equation we recognize that the second term should be color(red)(10^2=100 so we should add $20$
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$\Rightarrow y = - \left({\left(x\right)}^{2} - 2 \left(x\right) \left(10\right) + 80 \textcolor{red}{+ 20 - 20}\right)$
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$\Rightarrow y = - \left({\left(x\right)}^{2} - 2 \left(x\right) \left(10\right) + 100 - 20\right)$
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$\Rightarrow y = - \left({\left(x\right)}^{2} - 2 \left(x\right) \left(10\right) + 100\right) + 20$
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$\Rightarrow y = - \left({\left(x\right)}^{2} - 2 \left(x\right) \left(10\right) + {10}^{2}\right) + 20$
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$\Rightarrow y = - {\left(x - 10\right)}^{2} + 20$
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Therefore,$y = - {\left(x - 10\right)}^{2} + 20 \text{ }$ is the vertex form of the parabola where $\text{ } \left(10 , 20\right)$ is its vertex.

Apr 26, 2017

$y = - {\left(x - 10\right)}^{2} + 20$

#### Explanation:

To write a quadratic trinomial in vertex from you need to change:

$y = a {x}^{2} + b x + c \text{ }$ into the form $\text{ } y = p {\left(x + q\right)}^{2} + t$

In $y = p {\left(x + q\right)}^{2} + t$ the vertex is at $\left(- q , t\right)$

$y = - {x}^{2} + 20 x - 80 \text{ } \leftarrow$ make ${x}^{2}$ positive

$y = - \left[{x}^{2} - 20 x + 80\right] \text{ } \leftarrow$ complete the square

$y = - \left[{x}^{2} - 20 x \textcolor{b l u e}{+ 100 - 100} + 80\right] \text{ } \leftarrow \textcolor{b l u e}{+ {\left(\frac{b}{2}\right)}^{2} - {\left(\frac{b}{2}\right)}^{2}}$

$y = - \left[{\left(x - 10\right)}^{2} - 20\right]$

$y = - {\left(x - 10\right)}^{2} + 20$

This is now vertex form, giving the vertex as $\left(10 , 20\right)$