# How do you convert the following equation from standard to vertex form by completing the square: y=3x^2+12x+5?

Nov 15, 2016

#### Explanation:

The given equation is the equation of a parabola that opens upward (or downward). The vertex form of the equation of a parabola that opens upward (or downward) is:

$y = a {\left(x - h\right)}^{2} + k$

where $\left(h , k\right)$ is the vertex and "a" is the coefficient of the ${x}^{2}$ term.

Given: $y = 3 {x}^{2} + 12 x + 5$

$a = 3$, therefore, add 0 in the form $3 {h}^{2} - 3 {h}^{2}$ to the equation:

$y = 3 {x}^{2} + 12 x + 3 {h}^{2} - 3 {h}^{2} + 5$

Factor out 3 from the first 3 terms:

$y = 3 \left({x}^{2} + 4 x + {h}^{2}\right) - 3 {h}^{2} + 5$

Set the middle term in right side of the pattern, ${\left(x - h\right)}^{2} = {x}^{2} - 2 h x + {h}^{2}$, equal to the middle term in the equation:

$- 2 h x = 4 x$

Solve for h:

$h = - 2$

Substitute the left side of the pattern into the equation:

$y = 3 {\left(x - h\right)}^{2} - 3 {h}^{2} + 5$

Substitute -2 for h:

$y = 3 {\left(x - - 2\right)}^{2} - 3 {\left(- 2\right)}^{2} + 5$

Combine the constant terms:

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

The above is the vertex form.

The vertex can be read directly from the equation; it is at:

$\left(- 2 , - 7\right)$