Question

Using the completing-the-square method, rewrite f(x)=x²+10x+7 in vertex form.

A) f(x)=(x+5)² −18
B) f(x)=(x+10)² +25
C) f(x)=(x+5)² +7
D) f(x)=(x+5)²

Answer 1

The function f(x) = x² + 10x + 7 can be rewritten in vertex form as f(x) = (x + 5)² - 18 by completing the square, which is option A.

Step-by-step explanation:

To rewrite the function f(x) = x² + 10x + 7 in vertex form using the completing-the-square method, first, we need to create a perfect square trinomial. A perfect square trinomial is derived from the expression (x + a)², which expands to x² + 2ax + a². To form a perfect square with the x² + 10x part of f(x), we need to add and subtract the square of half the coefficient of x, which in this case is (10/2)² = 25.

So, we rewrite the function as:

f(x) = (x² + 10x + 25) - 25 + 7

Now, (x² + 10x + 25) is a perfect square and can be written as (x + 5)².

The function simplifies to:

f(x) = (x + 5)² - 18

Therefore, the correct vertex form of the function f(x) is f(x) = (x + 5)² - 18, which corresponds to option A.