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.

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)²to 3(x² - 4x) + 10. The way we are going to do that is by adding (-2)²and then removing (-2)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

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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