Solving Quadratic Equations by Factoring: A Step-by-Step Guide

Solving Quadratic Equations by Factoring: A Step-by-Step Guide 🧑‍🏫

Solving quadratic equations by factoring is a fundamental skill in algebra. 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 allows us to rewrite the quadratic expression as a product of two binomials, making it easier to find the solutions (also called roots or zeros) of the equation.

Step 1: Write the Quadratic Equation in Standard Form ✍️

Ensure the equation is in the standard form: $ax^2 + bx + c = 0$. This means all terms are on one side of the equation, and the other side is zero. If it's not, rearrange the terms to achieve this form.

Example:
2x^2 + 5x = 3  =>  2x^2 + 5x - 3 = 0

Step 2: Factor the Quadratic Expression ➗

The goal is to express the quadratic expression $ax^2 + bx + c$ as a product of two binomials $(px + q)(rx + s)$. There are several techniques for factoring, including:

• Trial and Error: Guess and check different combinations of factors.
• Factoring by Grouping: Useful when $a \neq 1$. Find two numbers that multiply to $ac$ and add up to $b$.

Let's illustrate factoring by grouping with an example:

Example:
Factor 2x^2 + 5x - 3

1. Find two numbers that multiply to (2)(-3) = -6 and add to 5. These numbers are 6 and -1.
2. Rewrite the middle term using these numbers: 2x^2 + 6x - x - 3
3. Group the terms: (2x^2 + 6x) + (-x - 3)
4. Factor out the greatest common factor (GCF) from each group: 2x(x + 3) - 1(x + 3)
5. Factor out the common binomial factor (x + 3): (2x - 1)(x + 3)

So, $2x^2 + 5x - 3 = (2x - 1)(x + 3)$.

Step 3: Set Each Factor Equal to Zero 0️⃣

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, set each factor equal to zero:

Example:
(2x - 1)(x + 3) = 0

Set each factor to zero:
2x - 1 = 0  or  x + 3 = 0

Step 4: Solve for x 🧮

Solve each equation obtained in the previous step to find the values of $x$:

Example:
2x - 1 = 0  =>  2x = 1  =>  x = 1/2
x + 3 = 0  =>  x = -3

Step 5: Write the Solutions ✅

The solutions to the quadratic equation are the values of $x$ found in the previous step. Write these solutions clearly.

Example:
The solutions to 2x^2 + 5x - 3 = 0 are x = 1/2 and x = -3.

Example Problem 1

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

1. Factor: $(x - 2)(x - 3) = 0$
2. Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$
3. Solve for $x$: $x = 2$ or $x = 3$

Solutions: $x = 2$ and $x = 3$.

Example Problem 2

Solve $x^2 - 9 = 0$.

1. Factor: $(x - 3)(x + 3) = 0$ (Difference of squares)
2. Set each factor to zero: $x - 3 = 0$ or $x + 3 = 0$
3. Solve for $x$: $x = 3$ or $x = -3$

Solutions: $x = 3$ and $x = -3$.

Summary 💡

Solving quadratic equations by factoring involves writing the equation in standard form, factoring the quadratic expression, setting each factor equal to zero, and solving for $x$. This method is efficient when the quadratic expression can be easily factored. Remember to always check your solutions by substituting them back into the original equation.