This is one way of writing a quadratic function:
[tex]f(x)= a(x-h)^2+k[/tex]
We're given:
[tex]f(x) = -6x^2 - 60x - 151[/tex]
To write a quadratic function in vertex form, we must complete the square.
⇒ Put parentheses around the first two terms containing x and x²:
[tex]f(x) = (-6x^2 - 60x) - 151[/tex]
⇒ Factor out -6 (keeping the x's in the parentheses):
[tex]f(x) = -6(x^2+10x) - 151[/tex]
⇒ To complete the square, add, inside the parentheses, the square of half the coefficient of x.
[tex]f(x) = -6(x^2+10x+25) - 151[/tex]
⇒ Now, we cannot randomly introduce a new value into a function. To balance the +25, subtract -6(25) outside the parentheses:
[tex]f(x) = -6(x^2+10x+25) - 151-(-6*25)\\f(x) = -6(x^2+10x+25) - 151-(-150)\\f(x) = -6(x^2+10x+25) - 151+150\\f(x) = -6(x^2+10x+25) - 1[/tex]
⇒ Finally, complete the square.
[tex]f(x) = -6(x+5)^2 - 1[/tex]
[tex]f(x) = -6(x+5)^2 - 1[/tex]
