What is the Factored Form of a Quadratic Equation?

The factored form of a quadratic equation is an expression written as a product of two linear factors.

A quadratic equation is generally represented in standard form as:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0.

The factored form expresses the quadratic equation as:

a(x - r₁)(x - r₂) = 0

where:

• a is the same leading coefficient as in the standard form.
• r₁ and r₂ are the roots or x-intercepts of the quadratic equation. These are the values of x that make the equation equal to zero.

Explanation:

The factored form highlights the roots of the quadratic equation. When either (x - r₁) or (x - r₂) equals zero, the entire expression becomes zero, satisfying the equation. Finding the roots is a key step in solving quadratic equations, and expressing the quadratic in factored form makes identifying these roots straightforward.

Example:

Consider the quadratic equation:

x² - 5x + 6 = 0

To find its factored form, we need to find two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Therefore, the factored form is:

(x - 2)(x - 3) = 0

In this example, the roots are x = 2 and x = 3.

Why is Factored Form Useful?

• Finding Roots: It directly reveals the roots or solutions of the quadratic equation.
• Solving Equations: It simplifies solving quadratic equations by setting each factor to zero.
• Graphing: The roots represent the x-intercepts of the parabola, which are crucial for graphing the quadratic function.