Final answer:
To write P(x) = -7x² + 21x in vertex form, complete the square to get P(x) = -7(x - 1.5)² + 15.75.
Step-by-step explanation:
To write the function P(x) = -7x² + 21x in the form P(x) = a(x - h)² + k, we need to complete the square. First, factor out the coefficient of the x² term from the x-terms.
P(x) = -7(x² - 3x)
Next, find the value that completes the square for the expression within the parentheses. This value is ¹⁄₂ the coefficient of x, squared.
(-3⁄₂)² = ¹⁄₄(3)² = ¹⁄₄(9) = ¹⁄₄(3²) = 2.25
Add and subtract this value inside the parentheses, then factor.
P(x) = -7[(x² - 3x + 2.25) - 2.25]
P(x) = -7[(x - 1.5)² - 2.25]
Distribute the -7 to the -2.25 inside the brackets.
P(x) = -7(x - 1.5)² + 15.75
So, the function in the form P(x) = a(x - h)² + k is P(x) = -7(x - 1.5)² + 15.75.