Question

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

Answer 1

The coordinates of the vertex are `(-5/2, 39/4)`.

Explanation:

`y=(x-3)(x-4)+4+12x`

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^2-7x+12+4+12x`

Now combine like terms.

`y=x^2+5x+16`

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

`y=x^2+5x+25/4+16-25/4`

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

`y=(x+5/2)^2+16-25/4`

Now combine the last two terms.

`y=(x+5/2)^2+39/4`

The equation is now in vertex form

`y=a(x-k)^2+h`

In this form, the coordinates of the vertex are `(k, h)`.

Here, `k=-5/2` and `h=39/4`, so the coordinates of the vertex are `(-5/2, 39/4)`.

Answer 2

The vertex is `(-5/2,39/4)` or `(-2.5,9.75)`.

Explanation:

Given:

`y=(x-3)(x-4)+4+12x`

First get the equation into standard form.

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

`y=x^2-7x+12+4+12x`

Collect like terms.

`y=x^2+(-7x+12x)+(12+4)`

Combine like terms.

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

`y=ax^2+bx+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=(-b)/(2a)`

`x=(-5)/(2*1)`

`x=-5/2=-2.5`

To find the `y` coordinate, substitute `-5/2` for `x` and solve for `y`.

`y=(-5/2)^2+5(-5/2)+16`

`y=25/4-25/2+16`

Multiply `25/2` and `16` by fractional forms of `1` to convert them to equivalent fractions with the denominator `4`.

`y=25/4-25/2xx2/2+16xx4/4`

`y=25/4-50/4+64/4`

`y=(25-50+64)/4`

`y=39/4=9.75`

The vertex is `(-5/2,39/4)` or `(-2.5,9.75)`.

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