Solving Quadratics by Completing the Square

Front

What is the method 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, making it easier to solve for x.

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 + 10x = -9?

Back

1. Move -9 to the other side: x² + 10x + 25 = 16. 2. Rewrite as (x + 5)² = 16. 3. Take the square root: x + 5 = ±4. 4. Solve for x: x = -1 or x = -9.

Front

What does the plus or minus symbol (±) indicate in solutions?

Back

The plus or minus symbol (±) indicates that there are two possible solutions: one with addition and one with subtraction.

Front

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

Back

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

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 discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root: b² - 4ac. It determines the nature of the roots.

Front

What does a positive discriminant indicate?

Back

A positive discriminant indicates that the quadratic equation has two distinct real roots.

Front

What does a zero discriminant indicate?

Back

A zero discriminant indicates that the quadratic equation has exactly one real root (a repeated root).

Front

What does a negative discriminant indicate?

Back

A negative discriminant indicates that the quadratic equation has no real roots, only complex roots.