Solving Quadratic Equations by Factoring

A quadratic equation is an equation that can be written in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are numbers, and $a \neq 0$. One of the most efficient ways to solve a quadratic equation is by factoring.

The Zero-Product Property

The entire method of solving by factoring relies on a simple logical rule called the Zero-Product Property. It states that if the product of two numbers (or expressions) is zero, then at least one of those numbers must be zero.

In math terms: If $a \cdot b = 0$, then $a = 0$ or $b = 0$.

Once we factor a quadratic equation into two binomials, we can set each binomial equal to zero to find our solutions.

Example 1: Simple Trinomials ($a = 1$)

Solve: $x^2 - 5x + 6 = 0$

1. Factor the quadratic: We need two numbers that multiply to $6$ (the constant term) and add up to $-5$ (the middle coefficient). Those numbers are $-2$ and $-3$.
2. Rewrite the equation: $(x - 2)(x - 3) = 0$
3. Apply the zero-product property: Set each factor to zero. $x - 2 = 0 \quad \text{or} \quad x - 3 = 0$
4. Solve for $x$: $x = 2 \quad \text{or} \quad x = 3$

Example 2: Complex Trinomials ($a \neq 1$)

Solve: $2x^2 + 7x - 15 = 0$

1. Factor by grouping: Multiply $a$ and $c$: $2 \times (-15) = -30$. We need two numbers that multiply to $-30$ and add to $7$. Those numbers are $10$ and $-3$.
2. Split the middle term: $2x^2 + 10x - 3x - 15 = 0$
3. Factor by grouping: $2x(x + 5) - 3(x + 5) = 0$ $(2x - 3)(x + 5) = 0$
4. Apply the zero-product property: $2x - 3 = 0 \quad \text{or} \quad x + 5 = 0$
5. Solve for $x$: $x = \frac{3}{2} \quad \text{or} \quad x = -5$

Example 3: Difference of Squares

Solve: $x^2 - 16 = 0$

1. Recognize the pattern: Both $x^2$ and $16$ are perfect squares. We can use the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$.
2. Factor the expression: $(x - 4)(x + 4) = 0$
3. Apply the zero-product property: $x - 4 = 0 \quad \text{or} \quad x + 4 = 0$
4. Solve for $x$: $x = 4 \quad \text{or} \quad x = -4$