Function in quadratic vertex form:

Question

A function is shown where b is a real number. f(x)=x^2+bx+144 The minimum value of the function is 80. Create an equation for an equivalent function in the form f(x)=(x-h)^2+k.

Answer 1

f'(x) = 2x + b = 0 when x = -b/2

80 = (-b/2)² + b(-b/2) + 144 = b²/4 - b²/2 + 144

-b²/4 = - 64 => b = 16

f(x) = x² + 16x + 144 = (x + 8)² + 80 (by completing the square)

Answer 2

Greetings! Lets solve this shall we !?

We have the polynomial f(x)=x²+bx+144, where b is a real number. What is b ? We can find out!

First, we find the factors of 144 where we can factor the polynomial and put it in quadratic vertex form (x-h)². Then, we notice that 12*12 = 144. Also, notice that the vertex formula has a negative on the inside. That means we must also find negative numbers (-12)(-12) to also equal 144. Now,

(x - 12)² = (x-12)(x-12) = x² - 12x - 12x + 144 = x² - 24x + 144. That means b = -24 and the vertex form is

f(x) = (x - 12)² + 0, where the vertex is (h, k) = (12, 0)............. k is zero because all we have is (x-12)².