Factoring and Solving Quadratic Equations: College Algebra Study Notes

Quadratic Equations and Factoring Methods

Introduction to Quadratic Equations

Quadratic equations are fundamental in College Algebra and appear in the form of polynomials of degree 2. Understanding their structure and how to solve them is essential for progressing in algebraic problem-solving.

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$.

• Standard Form: All terms are on one side, arranged in descending order of power.

• Example: $2x^2 - 6 = 0$

• Identifying Coefficients: For $ax^2 + bx + c = 0$, $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term.

Writing Quadratic Equations in Standard Form

To solve or factor quadratic equations, it is important to first write them in standard form.

• Standard Form: $ax^2 + bx + c = 0$

• Example 1: $5x^2 = x - 3$ becomes $5x^2 - x + 3 = 0$ Coefficients: $a = 5$, $b = -1$, $c = 3$

• Example 2: $-2x^2 + \frac{5}{3} = 0$ Coefficients: $a = -2$, $b = 0$, $c = \frac{5}{3}$

Factoring Quadratic Equations

Factoring is a key method for solving quadratic equations. The process involves expressing the quadratic as a product of two binomials and setting each factor equal to zero.

• Factoring Process:

  1. Write the equation in standard form.

  2. Factor completely.

  3. Set each factor equal to zero and solve for $x$.

  4. Check solutions in the original equation.

• Example: $x^2 - 9x = -20$ Standard Form: $x^2 - 9x + 20 = 0$ Factoring: $(x - 5)(x - 4) = 0$ Solutions: $x = 5$, $x = 4$

Factoring Methods: Decision Flowchart

Choosing the correct factoring method depends on the number of terms and the structure of the polynomial.

• Step 1: Factor out the Greatest Common Factor (GCF) if possible.

• Step 2: Count the number of terms:

  • 2 Terms: Use formulas for difference of squares, sum/difference of cubes.

  • 3 Terms: Check if it fits a factoring formula (e.g., perfect square trinomial). If not, use the AC method.

  • 4 Terms: Use grouping.

Factoring Formula	Expression
Difference of Squares	$(a + b)(a - b) = a^2 - b^2$
Sum of Cubes	$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
Difference of Cubes	$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
Factoring by Grouping	$(a - b)(a^2 + ab + b^2) = a^3 - b^3$

Solving Quadratic Equations by Factoring

Once a quadratic equation is factored, set each factor equal to zero to find the solutions (roots).

• Example: $x^2 - x - 6 = 0$ Factoring: $(x - 3)(x + 2) = 0$ Solutions: $x = 3$, $x = -2$

• Key Terms: Solutions, Roots, Zeros — all refer to values of $x$ that make the equation true.

Practice Problems

• Write the quadratic equation in standard form and identify $a$, $b$, $c$: $-4x^2 + x = 8$ Standard Form: $-4x^2 + x - 8 = 0$ $a = -4$, $b = 1$, $c = -8$

• Solve by factoring: $3x^2 + 12x = 0$ Factoring: $3x(x + 4) = 0$ Solutions: $x = 0$, $x = -4$

• Solve by factoring: $2x^2 + 7x + 6 = 0$ Factoring: $(2x + 3)(x + 2) = 0$ Solutions: $x = -\frac{3}{2}$, $x = -2$

Summary Table: Steps for Solving Quadratic Equations by Factoring

Step	Description
1	Write equation in standard form
2	Factor completely
3	Set each factor equal to zero and solve for $x$
4	Check solutions in the original equation

Additional info:

• Factoring is one of several methods for solving quadratic equations; others include completing the square and using the quadratic formula.

• Recognizing the structure of the quadratic equation helps in choosing the most efficient factoring method.