# What is the vertex of  y= -3x^2 - 2x - (x+2)^2 ?

May 11, 2017

The vertex is at $\left(- \frac{3}{4} , - \frac{7}{4}\right)$

#### Explanation:

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

Expand the polynomial:
$y = - 3 {x}^{2} - 2 x - \left({x}^{2} + 4 x + 4\right)$

Combine like terms:
$y = - 4 {x}^{2} - 6 x - 4$

Factor out $- 4$:
$y = - 4 \left[{x}^{2} + \frac{3}{2} x + 1\right]$

Complete the square:
$y = - 4 \left[{\left(x + \frac{3}{4}\right)}^{2} - {\left(\frac{3}{4}\right)}^{2} + 1\right]$

$y = - 4 \left[{\left(x + \frac{3}{4}\right)}^{2} + \frac{7}{16}\right]$

$y = - 4 {\left(x + \frac{3}{4}\right)}^{2} - \frac{7}{4}$

From vertex form, the vertex is at $\left(- \frac{3}{4} , - \frac{7}{4}\right)$

May 11, 2017

Vertex: $\left(- \frac{3}{4} , - \frac{55}{16}\right) \approx \left(- 0.75 , - 3.4375\right)$

#### Explanation:

1) Rewrite this equation in standard form
$y = - 3 {x}^{2} - 2 x - {\left(x + 2\right)}^{2}$
$y = - 3 {x}^{2} - 2 x - \left({x}^{2} + 4 x + 4\right)$
$y = - 4 {x}^{2} - 6 x - 4$

2) Rewrite this equation in vertex form by completing the square
$y = \left(- 4 {x}^{2} - 6 x\right) - 4$
$y = - 4 \left({x}^{2} + \frac{3}{2} x\right) - 4$
$y = - 4 \left({x}^{2} + \frac{3}{2} x + {\left(\frac{3}{4}\right)}^{2}\right) - 4 + {\left(\frac{3}{4}\right)}^{2}$
$y = - 4 {\left(x + \frac{3}{4}\right)}^{2} - \frac{55}{16}$

The vertex form is $y = a {\left(x - h\right)}^{2} + k$ reveals the vertex at $\left(h , k\right)$

Vertex: $\left(- \frac{3}{4} , - \frac{55}{16}\right) \approx \left(- 0.75 , - 3.4375\right)$

You can see this if you graph the equation
graph{y=-4x^2-6x-4 [-3, 2, -7, 0.1]}