Solve Quadratics by Factoring - Algebra-1

1. Fundamental Concepts

Solving quadratics by factoring is a method to find the solutions of a quadratic equation \(ax^2 + bx + c = 0\) (where \(a \neq 0\)) by rewriting the quadratic expression as a product of two binomial factors. It relies on the Zero Product Property (ZPP), which states that if the product of two factors is 0, then at least one of the factors must be 0. In mathematical terms: if \((mx + n)(px + q) = 0\), then \(mx + n = 0\) or \(px + q = 0\).

This method simplifies the quadratic equation into linear equations, making it easier to solve for the variable x.

Quadratic Equation Form: The equation must first be rearranged into the standard form \(ax^2 + bx + c = 0\) (all terms on one side, 0 on the other side).
Factoring Requirement: The quadratic expression \(ax^2 + bx + c\) must be factorable into two binomials (e.g., \((mx + n)(px + q)\)) with integer coefficients (in most basic cases).
Connection to ZPP: Factoring converts the quadratic equation into a product of factors, allowing the application of ZPP to find solutions.
Solutions as Roots: The solutions obtained are the roots of the quadratic equation, corresponding to the x-intercepts of the parabola \(y = ax^2 + bx + c\).

Equation: \(x^2 + 5x + 6 = 0\)

Steps:

Equation: \(2x^2 - 7x + 3 = 0\)

Steps:

Factor the quadratic expression (considering the coefficient of \(x^2\)):
- Find two numbers that multiply to \(2 \times 3 = 6\) and add to \(-7\): \(-1\) and \(-6\).
- Rewrite and factor by grouping: \(2x^2 - x - 6x + 3 = x(2x - 1) - 3(2x - 1) = (x - 3)(2x - 1)\).
Rewrite the equation: \((x - 3)(2x - 1) = 0\).
Apply ZPP: \(x - 3 = 0\) or \(2x - 1 = 0\).
Solve for x: \(x = 3\) or \(x = \frac{1}{2}\).

Equation: \(3x^2 + 5x - 12 = 0\)

Steps:

Factor the quadratic expression (with a non-1 coefficient for \(x^2\)):
- Find two numbers that multiply to \(3 \times (-12) = -36\) and add to 5: 9 and \(-4\).
- Rewrite and factor by grouping: \(3x^2 + 9x - 4x - 12 = 3x(x + 3) - 4(x + 3) = (3x - 4)(x + 3)\).
Rewrite the equation: \((3x - 4)(x + 3) = 0\).
Apply ZPP: \(3x - 4 = 0\) or \(x + 3 = 0\).
Solve for x: \(x = \frac{4}{3}\) or \(x = -3\).

Write the equation in standard form: Ensure all terms are on one side, so the equation is \(ax^2 + bx + c = 0\).
Factor out the greatest common factor (GCF) first: If the coefficients have a common factor, factor it out to simplify the quadratic expression (e.g., \(3x^2 + 6x = 0\) becomes \(3x(x + 2) = 0\)).
Factor the simplified quadratic:
- For \(a = 1\): Find two numbers that multiply to c and add to b, then write as \((x + m)(x + n)\).
- For \(a \neq 1\): Use the "AC method" (multiply a and c, find two numbers with a product of ac and sum of b, then factor by grouping).
Apply the Zero Product Property: Set each factor equal to 0.
Solve the linear equations: Find the values of x that satisfy each factor being 0.
Check solutions (optional): Substitute the solutions back into the original equation to verify they make the equation true.