# What is the the vertex of y = (x-3)(x-4) +4+12x ?

Apr 4, 2018

The coordinates of the vertex are $\left(- \frac{5}{2} , \frac{39}{4}\right)$.

#### Explanation:

$y = \left(x - 3\right) \left(x - 4\right) + 4 + 12 x$

Let's put this in standard form first. Expand the first term on the right-hand side using the distributive property (or FOIL if you like).

$y = {x}^{2} - 7 x + 12 + 4 + 12 x$

Now combine like terms.

$y = {x}^{2} + 5 x + 16$

Now complete the square by adding and subtracting (5/2)^2 to the right-hand side.

$y = {x}^{2} + 5 x + \frac{25}{4} + 16 - \frac{25}{4}$

Now factor the first three terms of the right-hand side.

$y = {\left(x + \frac{5}{2}\right)}^{2} + 16 - \frac{25}{4}$

Now combine the last two terms.

$y = {\left(x + \frac{5}{2}\right)}^{2} + \frac{39}{4}$

The equation is now in vertex form

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

In this form, the coordinates of the vertex are $\left(k , h\right)$.

Here, $k = - \frac{5}{2}$ and $h = \frac{39}{4}$, so the coordinates of the vertex are $\left(- \frac{5}{2} , \frac{39}{4}\right)$.

Apr 5, 2018

The vertex is $\left(- \frac{5}{2} , \frac{39}{4}\right)$ or $\left(- 2.5 , 9.75\right)$.

#### Explanation:

Given:

$y = \left(x - 3\right) \left(x - 4\right) + 4 + 12 x$

First get the equation into standard form.

FOIL $\left(x - 3\right) \left(x - 4\right)$.
https://www.ipracticemath.com/learn/algebra/foil-method-of-binomial-multiplication

$y = {x}^{2} - 7 x + 12 + 4 + 12 x$

Collect like terms.

$y = {x}^{2} + \left(- 7 x + 12 x\right) + \left(12 + 4\right)$

Combine like terms.

color(blue)(y=x^2+5x+16 is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c$,

where:

$a = 1$, $b = 5$, $c = 16$

The vertex is the maximum or minimum point of a parabola. The $x$ coordinate can be determined by using the formula:

$x = \frac{- b}{2 a}$

$x = \frac{- 5}{2 \cdot 1}$

$x = - \frac{5}{2} = - 2.5$

To find the $y$ coordinate, substitute $- \frac{5}{2}$ for $x$ and solve for $y$.

$y = {\left(- \frac{5}{2}\right)}^{2} + 5 \left(- \frac{5}{2}\right) + 16$

$y = \frac{25}{4} - \frac{25}{2} + 16$

Multiply $\frac{25}{2}$ and $16$ by fractional forms of $1$ to convert them to equivalent fractions with the denominator $4$.

$y = \frac{25}{4} - \frac{25}{2} \times \frac{2}{2} + 16 \times \frac{4}{4}$

$y = \frac{25}{4} - \frac{50}{4} + \frac{64}{4}$

$y = \frac{25 - 50 + 64}{4}$

$y = \frac{39}{4} = 9.75$

The vertex is $\left(- \frac{5}{2} , \frac{39}{4}\right)$ or $\left(- 2.5 , 9.75\right)$.

graph{y=x^2+5x+16 [-13.5, 11.81, 6.47, 19.12]}