Functions
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Factoring and Roots

See where (x - a)(x - b) crosses zero — and why factoring reveals the answer

Why do we factor? Because finding roots — where an expression equals zero — is how we solve equations, find where curves cross the x-axis, and answer real-world questions like "when does this projectile hit the ground?" When you factor an expression like x^2 - x - 2 into (x - 2)(x + 1), you're doing more than rearranging symbols — you're revealing the roots, the values of x where the expression equals zero.

The zero product property says: if two things multiply to zero, at least one of them must be zero. So if (x - 2)(x + 1) = 0, then either x - 2 = 0 (giving x = 2) or x + 1 = 0 (giving x = -1). The roots jump right out of the factored form.

In this lesson, you'll see this on a graph: the curve y = (x - a)(x - b) crosses the x-axis at exactly x = a and x = b. Drag the sliders to move the roots around and watch the curve reshape itself — with an AI tutor explaining every step.

Graph

FAQ

What is factoring?
Factoring means rewriting an expression as a product of simpler pieces. For example, x^2 - x - 2 = (x - 2)(x + 1). The "factors" are (x - 2) and (x + 1). Factoring is the reverse of expanding — instead of multiplying out, you're breaking down.
How do I find the roots of a quadratic?
If you can factor the quadratic, the roots come directly from the factors. For (x - 2)(x + 1) = 0, set each factor to zero: x - 2 = 0 gives x = 2, and x + 1 = 0 gives x = -1. If you can't factor easily, use the quadratic formula.
What is the zero product property?
The zero product property states: if A × B = 0, then A = 0 or B = 0 (or both). This is why factoring works for finding roots — once you write the expression as a product equal to zero, each factor gives you a root.
What is the difference between factored form and standard form?
Factored form is y = (x - a)(x - b) — you can read the roots directly (x = a and x = b). Standard form is y = x^2 + bx + c — the expanded version. Both describe the same curve; factored form reveals the roots, standard form reveals the coefficients.

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Quadratic EquationsPowers and ExponentsWhat is a Function?