Solving Quadratic Equations by Completing the Square

Front

What is the process of completing the square in a quadratic equation?

Back

Completing the square involves rewriting a quadratic equation in the form (x + p)² = q, where p and q are constants. This method helps to solve the equation by making it easier to find the roots.

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 = 0?

Back

1. Move the constant to the other side: x² + 6x = 5. 2. Take half of the coefficient of x (which is 6), square it (3² = 9), and add it to both sides: x² + 6x + 9 = 14. 3. Factor the left side: (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 does it mean to factor a quadratic equation?

Back

Factoring a quadratic equation means expressing it as a product of two binomials, such as (x - p)(x - q) = 0, where p and q are the roots of the equation.

Front

How do you find the roots of a quadratic equation using completing the square?

Back

1. Rewrite the equation in the form (x + p)² = q. 2. Take the square root of both sides. 3. Solve for x by isolating it.

Front

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

Back

The discriminant (b² - 4ac) determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root; if it's negative, there are two complex roots.

Front

How do you solve x² - 4x = 16 by completing the square?

Back

1. Move 16 to the left: x² - 4x - 16 = 0. 2. Complete the square: (x - 2)² = 20.

Front

What is the first step in completing the square for x² + 4x = 12?

Back

Move the constant to the other side: x² + 4x - 12 = 0.

Front

What is the formula for completing the square?

Back

To complete the square for ax² + bx, use the formula: (b/2a)².