Solve Quadratic Equations by Factoring - Algebra-2

1. Fundamental Concepts

Definition: A quadratic equation is an equation of the form $$ax^2 + bx + c = 0\$$ , where $$a\$$ , $$b\$$ , and $$c\$$ are constants, and $$a \neq 0\$$ .
Factoring: Factoring a quadratic equation involves expressing it as a product of two binomials, $$ (px + q)(rx + s) = 0 \$$ , where $$p, q, r,\$$ and $$s\$$ are constants.
Zero Product Property: If the product of two factors is zero, then at least one of the factors must be zero. This property is used to solve factored quadratic equations.

Standard Form: $$ax^2 + bx + c = 0\$$

Factored Form: $$ (px + q)(rx + s) = 0 \$$

Solution Method: Set each factor equal to zero and solve for $$x\$$ .

Problem: Solve $$x^2 - 5x + 6 = 0\$$ by factoring.

Factor the quadratic expression: $$x^2 - 5x + 6 = (x - 2)(x - 3)\$$
Set each factor equal to zero:
- $$x - 2 = 0 \Rightarrow x = 2\$$
- $$x - 3 = 0 \Rightarrow x = 3\$$

Validation: Substitute $$x = 2\$$ and $$x = 3\$$ into the original equation:

Problem: Solve $$2x^2 + 7x + 3 = 0\$$ by factoring.

Factor the quadratic expression: $$2x^2 + 7x + 3 = (2x + 1)(x + 3)\$$
Set each factor equal to zero:
- $$2x + 1 = 0 \Rightarrow x = -\frac{1}{2}\$$
- $$x + 3 = 0 \Rightarrow x = -3\$$

Validation: Substitute $$x = -\frac{1}{2}\$$ and $$x = -3\$$ into the original equation:

$$2\left(-\frac{1}{2}\right)^2 + 7\left(-\frac{1}{2}\right) + 3 = 2\left(\frac{1}{4}\right) - \frac{7}{2} + 3 = \frac{1}{2} - \frac{7}{2} + 3 = 0\$$ ✓
$$2(-3)^2 + 7(-3) + 3 = 2(9) - 21 + 3 = 18 - 21 + 3 = 0\$$ ✓

Identify Coefficients: Clearly identify the values of $$a\$$ , $$b\$$ , and $$c\$$ in the standard form $$ax^2 + bx + c = 0\$$ .
Find Factors: Look for pairs of numbers that multiply to $$ac\$$ and add to $$b\$$ .
Check Solutions: Always verify the solutions by substituting them back into the original equation.
Use Visual Aids: Draw a table or use a factor tree to help find the correct pair of factors.