5.5 Factoring Quadratic Functions

Front

What is a quadratic function?

Back

A quadratic function is a polynomial function of degree 2, typically written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Front

What does it mean to factor a quadratic function?

Back

Factoring a quadratic function involves expressing it as a product of two binomials, such as (x + p)(x + q), where p and q are the roots of the function.

Front

What is the standard form of a quadratic function?

Back

The standard form of a quadratic function is f(x) = ax² + bx + c.

Front

What is the factored form of a quadratic function?

Back

The factored form of a quadratic function is f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots of the function.

Front

How do you find the roots of a quadratic function?

Back

The roots of a quadratic function can be found using the quadratic formula: r = (-b ± √(b² - 4ac)) / (2a), or by factoring the quadratic expression.

Front

What is the quadratic formula?

Back

The quadratic formula is r = (-b ± √(b² - 4ac)) / (2a), used to find the roots of a quadratic equation.

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; if negative, two complex roots.

Front

What are the steps to factor a quadratic expression?

Back

1. Write the expression in standard form. 2. Identify a, b, and c. 3. Find two numbers that multiply to ac and add to b. 4. Rewrite the middle term using these numbers. 5. Factor by grouping.

Front

What is the significance of the leading coefficient in a quadratic function?

Back

The leading coefficient (a) determines the direction of the parabola: if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

Front

What is a binomial?

Back

A binomial is a polynomial with two terms, such as (x + 3) or (2x - 5).