Factoring is a powerful way to solve quadratic equations because it turns one curved equation into two simpler linear equations. A quadratic equation has the form ax² + bx + c = 0, and its solutions are called roots or zeros. On a graph, these solutions are the x-values where the parabola crosses or touches the x-axis.
This method matters because it connects algebraic structure with visible features of a graph.
The key idea is the zero product property: if two factors multiply to make 0, then at least one factor must be 0. After rewriting a quadratic as a product, such as (x - r₁)(x - r₂) = 0, each factor can be set equal to 0 and solved. Factoring works best when the quadratic can be written using integer or simple rational factors.
It is often the fastest method when the equation is already factorable or close to factored form.
Key Facts
- Standard form of a quadratic equation: ax² + bx + c = 0, where a ≠ 0.
- Zero product property: If AB = 0, then A = 0 or B = 0.
- Factored form with roots r₁ and r₂: a(x - r₁)(x - r₂) = 0.
- If (x - r₁)(x - r₂) = 0, then x = r₁ or x = r₂.
- For x² + bx + c, find numbers m and n such that m + n = b and mn = c.
- The roots of ax² + bx + c = 0 are the x-intercepts of y = ax² + bx + c.
Vocabulary
- Quadratic equation
- An equation whose highest power of the variable is 2, usually written as ax² + bx + c = 0.
- Factor
- An expression that is multiplied by another expression to make a product.
- Zero product property
- The rule that if a product equals 0, then at least one of its factors must equal 0.
- Root
- A solution to an equation, or an x-value that makes the equation equal 0.
- x-intercept
- A point where a graph crosses or touches the x-axis, so its y-value is 0.
Common Mistakes to Avoid
- Forgetting to set the quadratic equal to 0. Factoring only solves the equation when the product is equal to 0, so first rewrite it in standard form.
- Factoring with the wrong signs. Check both the product and the sum of the factor numbers because sign errors often give the wrong middle term.
- Stopping after factoring and not solving each factor. A factored expression like (x - 3)(x + 5) = 0 must become x - 3 = 0 and x + 5 = 0.
- Assuming every quadratic has two different x-intercepts. A quadratic can have two, one, or no real x-intercepts depending on its roots.
Practice Questions
- 1 Solve by factoring: x² - 7x + 12 = 0.
- 2 Solve by factoring: 2x² + 5x - 3 = 0.
- 3 Explain how the zero product property connects the factored form of a quadratic to the x-intercepts of its graph.