# What is the vertex form of y= x^2-3x-28 ?

Feb 1, 2016

color(blue)"Shortcut method - by sight")

Given$\to y = {x}^{2} - 3 x - 28$ .......................................(1)

$y = {\left(x - \frac{3}{2}\right)}^{2} - \frac{3}{4} - 28$

$y = {\left(x - \frac{3}{2}\right)}^{2} - \frac{121}{4}$

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$\textcolor{p u r p \le}{\text{Fuller explanation}}$

$\textcolor{b l u e}{\text{Step 1}}$

Write as$\text{ } y = \left({x}^{2} - 3 x\right) - 28$

$\textcolor{b r o w n}{\text{Divide the brackets contents by "x". These means that the right}}$$\textcolor{b r o w n}{\text{hand side is no longer equal to } y}$

$y \ne \left(x - 3\right) - 28$

$\textcolor{b r o w n}{\text{square the brackets}}$

$y \ne {\left(x - 3\right)}^{2} - 28$

$\textcolor{b r o w n}{\text{Halve the -3 from } \left(x - 3\right)}$

$y \ne {\left(x - \frac{3}{2}\right)}^{2} - 28$
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$\textcolor{b l u e}{\text{Step 2}}$

$\textcolor{b r o w n}{\text{Changing the equation so that it does equal } y}$

Let a constant of correction be k then

$y = {\left(x - \frac{3}{2}\right)}^{2} - 28 + k$...................................(2)

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$\textcolor{b l u e}{\text{Step 3}}$

$\textcolor{b r o w n}{\text{To find the value of k}}$

$\textcolor{g r e e n}{\text{As equation (1) and equation (2) both equal y we can equate them}}$ $\textcolor{g r e e n}{\text{to each other through y}}$

Equation (1) = y = Equation (2)

${x}^{2} - 3 x - 28 \text{ "=" } {\left(x - \frac{3}{2}\right)}^{2} - 28 + k$

$\cancel{{x}^{2}} - \cancel{3 x} - \cancel{28} \text{ "=" } \cancel{{x}^{2}} - \cancel{3 x} + \frac{9}{4} - \cancel{28} + k$

$k = - \frac{9}{4}$......................................................(3)
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$\textcolor{b l u e}{\text{Step 4 - last move!}}$

$\textcolor{b r o w n}{\text{Bringing it all together to give the final equation}}$

Substitute equation (3) into equation (2)

$y = {\left(x - \frac{3}{2}\right)}^{2} - 28 - \frac{9}{4}$.

But $- 28 - \frac{9}{4} = - \frac{121}{4}$ giving

color(green)(y=(x-3/2)^2-121/4.