Convert a quadratic function from the general form to the vertex form

We show you in this lesson two ways to convert a quadratic function from the general form to the vertex form. 

How to convert a quadratic function from the general form to the vertex form by completing the square.

Write the function f(x) = 3x2 - 12x + 10 in vertex form by completing the square

f(x) = 3x2 - 12x + 10

First, group the quadratic term and the linear term.

f(x) = (3x2 - 12x) + 10

Since we need a leading coefficient of 1 to complete the square, factor out the 3.

f(x) = 3(x2 - 4x) + 10

Complete the square. Review this lesson about completing the square if needed.
Basically, you need to find the constant term for x2 - 4x.

The constant term is (-4/2)2 = (-2)2 

Add (-2)to 3(x2 - 4x) + 10. The way we are going to do that is by adding (-2)and then removing (-2) so we do not change the function.

f(x) = 3(x2 - 4x + (-2)2 - (-2)2) + 10

f(x) = 3(x2 - 4x + (-2)2 - 4) + 10

f(x) = 3(x2 - 4x + (-2)2)+ 3(- 4) + 10

f(x) = 3(x2 - 4x + (-2)2) + -12 + 10

f(x) = 3(x - 2)2 + -2

The vertex form is f(x) = 3(x - 2)2 + -2

How to convert a quadratic function from the general form to the vertex form by using the formula to find the x-coordinate of the vertex: x = -b/2a

Write the function f(x) = 3x2 - 12x + 10 in vertex form using the formula to find the x-coordinate of the vertex.

f(x) = 3x2 - 12x + 10

Find the x-coordinate of the vertex using x = -b/2a

a = 3, b = -12 and c = 10

x = - -12/2(3) = 12/6 = 2

Find the y-coordinate of the vertex using f(x) = 3x2 - 12x + 10

f(2) = 3(2)2 - 12(2) + 10

f(2) = 3(4) - 24 + 10

f(2) = 12 - 24 + 10

f(2) = -12 + 10

f(2) = -2

The vertex form of a quadratic function is f(x) = a(x - h)2 + k, where a, h, and k are real numbers and (h,k) is the vertex.

Therefore, a  = 3, h = 2 and k  = -2

f(x) = 3(x - 2)2 + -2

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