SOLUTION: How to get the quadratic Intercept form converted to the quadratic vertex form in the equation y=4(x+1)(x-6)

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Question 238027: How to get the quadratic Intercept form converted to the quadratic vertex form in the equation y=4(x+1)(x-6)
Answer by Theo(13342) About Me  (Show Source):
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graph%28400%2C400%2C-10%2C10%2C-50%2C50%2C4x%5E2+-+20x+-+24%29

The above is a graph of your equation in standard form of y = 4x^2 - 20x - 24.

The minimum point of the graph is at (2.5,-49).

The intercept form of this equation is y = (x+1)*(x-6)

There are a couple of ways to get to the vertex form of this equation.

Both involve converting to the standard form of the equation first.

The standard form is ax^2 + bx + c

Simply multiply your factors out to get:

y = 4x^2 - 20x - 24 = 0

The first way is to solve for the x value of the max/min point which is the x value of the vertex.

The formula for that is x = -b/2a

That becomes 20/8 = 2.5

Then you want to find the y point.

The y point is f(-b/2a) which becomes f(2.5) which means you replace x in the standard form of the equation and solve for y.

Your equation of y = 4x^2 - 20x - 24 becomes:

y = 4*(2.5)^2 - 20*(2.5) - 24 which becomes:

y = -49.

The vertex of your equation is (2.5,-49)

The vertex form of your equation is:

y = a * (x-h)^2 + k where (h,k) is the vertex of the equation.

This standard form would become:

y = a *(x-2.5)^2 - 49

It is left only to solve for a.

The a is the a term in the standard form of the equation which is 4.

The vertex form of your equation is then:

y = 4*(x-2.5)^2 - 49

graph of this equation is shown below:

graph%28400%2C400%2C-10%2C10%2C-50%2C50%2C4%2A%28x-2.5%29%5E2+-+49%29

the graph confirms the standard form of the equation and the vertex form of the equation is identical.

The second way to convert to the vertex form is to get the standard form and then use the completing the square method to convert to the vertex form.

That is done as follows:

The standard form of your equation is:

y = 4x^2 - 20x - 24 = 0

add 24 to both sides to get:

4x^2 - 20x = 24

divid both sides of your equation by 4 which is the a term.

hold on to the a term however because you will be multiplying it back in after you're done.

you get:
x^2 - 5x = 6

take 1/2 of the b term to get 2.5
square 1/2 of the b term to get 2.5^2 = 6.25

add 6.25 to the right side of your equation to get:

x^2 - 5x = 12.25

replace your b term on the left side of the equation with 1/2 of the b term and than divide the expression by x and then square it to get:

(x-2.5)^2 = 12.25

now multiply both sides of your equation by the a term of 4 that you divided out earlier, to get:

4 * (x-2.5)^2 = 49

subtract 49 from both sides of this equation to get:

4 * (x-2.5)^2 - 49 = 0

that's your vertex form.

standard form is y = a * (x-h)^2 + k

h = 2.5
k = -4.9

vertex is equal to (h,k) = (2.5,-49) as seen on the graph.