Which technique converts a quadratic equation from standard form into perfect square form?

Which technique converts a quadratic equation from standard form into perfect square form?

• A. Completing the square
• B. Factoring
• C. Using the quadratic formula
• D. Graphing

Answer: (A)

Completing the square is the technique used to rewrite a quadratic in standard form as a perfect square form. The idea is to form a square of a binomial plus or minus a constant by adding and subtracting just the right amount. For a quadratic ax^2 + bx + c, start by factoring out a from the x-terms: a[x^2 + (b/a)x] + c. Inside the brackets, add and subtract (b/(2a))^2 to create a perfect square: a[x^2 + (b/a)x + (b/(2a))^2] - a(b/(2a))^2 + c which simplifies to a(x + b/(2a))^2 + c - b^2/(4a). When a = 1, this becomes x^2 + bx + c = (x + b/2)^2 + (c - b^2/4). Example: rewrite x^2 + 6x + 5 as a perfect square form. Half of 6 is 3, and 3^2 is 9, so x^2 + 6x + 9 - 9 + 5 = (x + 3)^2 - 4. Thus the perfect square form is (x + 3)^2 - 4. This method is specifically about turning the quadratic into a square-plus-constant form; other techniques like factoring, the quadratic formula, or graphing serve different purposes.

Completing the square is the technique used to rewrite a quadratic in standard form as a perfect square form. The idea is to form a square of a binomial plus or minus a constant by adding and subtracting just the right amount.

For a quadratic ax^2 + bx + c, start by factoring out a from the x-terms: a[x^2 + (b/a)x] + c. Inside the brackets, add and subtract (b/(2a))^2 to create a perfect square:

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

which simplifies to a(x + b/(2a))^2 + c - b^2/(4a).

When a = 1, this becomes x^2 + bx + c = (x + b/2)^2 + (c - b^2/4).

Example: rewrite x^2 + 6x + 5 as a perfect square form.

Half of 6 is 3, and 3^2 is 9, so

x^2 + 6x + 9 - 9 + 5 = (x + 3)^2 - 4.

Thus the perfect square form is (x + 3)^2 - 4.

This method is specifically about turning the quadratic into a square-plus-constant form; other techniques like factoring, the quadratic formula, or graphing serve different purposes.