Question

What does it mean to factor a quadratic expression?

Short answer

Factoring a quadratic expression means breaking it down into simpler binomial expressions that, when multiplied together, result in the original quadratic expression. For example, \(x^2 + 5x + 6\) can be factored into \((x+2)(x+3)\).

Step-by-step solution

Understand the structure of a quadratic expression

A quadratic expression typically takes the form \(ax^2 + bx + c\), where a, b, and c are constants. Recognizing this structure is the first step toward being able to factor the expression. However, it should be noted that the coefficients do not need to be numerical, but can be any constant expression.

Identify the goal

Factoring a quadratic expression means breaking it down into simpler binomial expressions that, when multiplied together, result in the original quadratic expression. This is, in essence, decomposition.

Factoring process

Start by looking for two numbers that multiply to give ac (the product of a and c) and add to give b. These numbers replace the middle term. Then, factor by grouping. Lastly, rewrite the expression as the product of two binomial expressions. For example, the quadratic expression \(x^2 + 5x + 6\) can be factored into \((x+2)(x+3)\).