Express 2x^2-4x-16 in the form a(x+h)^2+k, where a,h,k are real numbers.
hmmm, maybe complete the square ?
2 x^2 - 4 x - 16 = y
x^2 - 2 x - 8 = y/2
x^2 - 2 x = y/2 + 8
x^2 - 2 x + 1 = y/2 + 9
(x+1)^2 = y/2 + 9
2 (x+1)^2 - 18 = y , done
gotta watch those +/- signs
y = 2x^2-4x-16
y = 2(x^2-2x+1) - 18
y = 2(x-1)^2 - 18
whoops sorry !
To express the quadratic expression 2x^2 - 4x - 16 in the form a(x + h)^2 + k, where a, h, and k are real numbers, we need to complete the square. Here's how to do it step by step:
Step 1: Start with the given quadratic expression.
f(x) = 2x^2 - 4x - 16
Step 2: Factor out the coefficient of x^2 from the first two terms.
f(x) = 2(x^2 - 2x) - 16
Step 3: Take half of the coefficient of x, square it, and add it inside the parentheses. Subtract the same value multiplied by the coefficient of x^2 outside the parentheses. This step is called completing the square.
f(x) = 2(x^2 - 2x + (-2/2)^2) - 16 - 2(-2/2)^2
Simplifying the equation:
f(x) = 2(x^2 - 2x + 1) - 16 - 2(1)
f(x) = 2(x - 1)^2 - 16 - 2
f(x) = 2(x - 1)^2 - 18
Step 4: Compare the resulting equation with the form a(x + h)^2 + k.
From the equation, a = 2, h = -1, and k = -18.
Therefore, 2x^2 - 4x - 16 can be expressed in the form a(x + h)^2 + k as 2(x - 1)^2 - 18, where a = 2, h = -1, and k = -18.