When completing the square for ax^2 + bx + c, which expression is added and subtracted to form the square term after factoring out a?

When completing the square for ax^2 + bx + c, which expression is added and subtracted to form the square term after factoring out a?

• A. (b/(2a))^2
• B. (b/a)^2
• C. (2a/b)^2
• D. (b/(2a))^3

Answer: (A)

When completing the square for ax^2 + bx + c, you first factor out a from the x^2 and x terms to get a(x^2 + (b/a)x) + c. To turn the inside into a perfect square, you add the square of half the coefficient of x inside the parentheses. Half of (b/a) is b/(2a), and its square is (b/(2a))^2. So you add and subtract (b/(2a))^2 inside the brackets: a[x^2 + (b/a)x + (b/(2a))^2] − a[(b/(2a))^2] + c The first part becomes a(x + b/(2a))^2, and the remaining constants combine to c − b^2/(4a). Therefore, the expression you add and subtract to form the square term is (b/(2a))^2.

When completing the square for ax^2 + bx + c, you first factor out a from the x^2 and x terms to get a(x^2 + (b/a)x) + c. To turn the inside into a perfect square, you add the square of half the coefficient of x inside the parentheses. Half of (b/a) is b/(2a), and its square is (b/(2a))^2. So you add and subtract (b/(2a))^2 inside the brackets:

a[x^2 + (b/a)x + (b/(2a))^2] − a[(b/(2a))^2] + c

The first part becomes a(x + b/(2a))^2, and the remaining constants combine to c − b^2/(4a). Therefore, the expression you add and subtract to form the square term is (b/(2a))^2.