# How do you find the coordinates of the vertex y= 2x^2 + 7x - 21 ?

Apr 4, 2017

Vertex is $\left(- \frac{7}{4} , - \frac{217}{8}\right)$ or $\left(- 1 \frac{3}{4} , - 27 \frac{1}{8}\right)$

#### Explanation:

To find the coordinates of vertex of $y = 2 {x}^{2} + 7 x - 21$, one should convert this equation into vertex form i.e.

$\left(y - k\right) = a {\left(x - h\right)}^{2}$, where vertex is $\left(h , k\right)$

Now $y = 2 {x}^{2} + 7 x - 21$

$\Leftrightarrow y = 2 \left({x}^{2} + \frac{7}{2} x\right) - 21$

$= 2 \left({x}^{2} + 2 \times \frac{7}{4} \times x + {\left(\frac{7}{4}\right)}^{2} - {\left(\frac{7}{4}\right)}^{2}\right) - 21$

$= 2 \left({\left(x + \frac{7}{4}\right)}^{2} - {\left(\frac{7}{4}\right)}^{2}\right) - 21$

$= 2 {\left(x + \frac{7}{4}\right)}^{2} - 2 \times \frac{49}{16} - 21$

$= 2 {\left(x + \frac{7}{4}\right)}^{2} - \frac{49}{8} - 21$

$= 2 {\left(x + \frac{7}{4}\right)}^{2} - \frac{217}{8}$

or $\left(y + \frac{217}{8}\right) = 2 {\left(x + \frac{7}{4}\right)}^{2}$

Hence, vertex is $\left(- \frac{7}{4} , - \frac{217}{8}\right)$ or $\left(- 1 \frac{3}{4} , - 27 \frac{1}{8}\right)$

graph{2x^2+7x-21 [-6, 4, -28.56, -8.56]}