3.5 Solving Quadratics by Completing the Square

Front

What is the process of completing the square?

Back

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form (x - p)² = q, where p and q are constants.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

How do you complete the square for the equation x² + 6x = 5?

Back

Add (6/2)² = 9 to both sides to get (x + 3)² = 14.

Front

What is the vertex form of a quadratic equation?

Back

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

What is the value to add to complete the square for x² - 4x?

Back

Add (4/2)² = 4 to complete the square, resulting in (x - 2)².

Front

How do you solve the equation x² + 6x + 9 = 0?

Back

Factor to (x + 3)² = 0, giving the solution x = -3.

Front

What is the significance of the discriminant in a quadratic equation?

Back

The discriminant (b² - 4ac) determines the nature of the roots: if > 0, two real roots; if = 0, one real root; if < 0, no real roots.

Front

How do you derive the quadratic formula from completing the square?

Back

Starting from ax² + bx + c = 0, divide by a, complete the square, and solve for x to derive x = (-b ± √(b² - 4ac)) / (2a).

Front

What is the first step in completing the square for x² + 14x?

Back

Take half of 14, which is 7, and square it to get 49, then add it to both sides.

Front

What does it mean to factor a quadratic equation?

Back

Factoring a quadratic equation means rewriting it as a product of two binomials, such as (x - p)(x - q) = 0.