Question

Use completing the square to rewrite the quadratic function into the form y= a(x + h)² + k.

f(x) = 4x² + 62x – 3

a) f(x) = (x + 2)² – 21
b) f(x) = (2x + 3)² - 2
c) f(x) = 4(x + 3) - 2
d) f(x) = 4(x + 12)² – 21

Answer 1

To complete the square for f(x) = 4x² + 62x − 3, we first factor out the coefficient of x² from the first two terms, add and subtract the square of half the coefficient of x inside the parentheses, then simplify and distribute to rewrite the function in vertex form.

Step-by-step explanation:

To rewrite the quadratic function f(x) = 4x² + 62x − 3 into the form y= a(x + h)² + k using completing the square, we start by factoring the coefficient of x² out of the first two terms:

f(x) = 4(x² + 15.5x) - 3

Now, we complete the square for the expression inside the parentheses. First, we need to add and subtract the square of half the x's coefficient inside the bracket, which is (15.5/2)²:

f(x) = 4[(x² + 15.5x + (15.5/2)²) - (15.5/2)²] - 3

Simplify the expression inside the brackets:

f(x) = 4[(x + 7.75)² - 60.0625] - 3

Distribute the 4:

f(x) = 4(x + 7.75)² - 240.25 - 3

Combine like terms:

f(x) = 4(x + 7.75)² - 243.25

So, the function rewritten in vertex form is:

f(x) = 4(x + 7.75)² - 243.25