Solving Quadratic Equations by Factoring

Quadratic equations are equations that can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations are often used to model real-world situations, such as the trajectory of a projectile or the growth of a population. Solving quadratic equations can be done in a variety of ways, but one of the most common methods is factoring.

What is Factoring?

Factoring is the process of breaking down a polynomial into a product of two or more simpler polynomials. In the context of quadratic equations, factoring involves finding two binomials that multiply together to give the original quadratic equation.

How to Solve Quadratic Equations by Factoring

Here are the steps to solve quadratic equations by factoring:

1. Write the quadratic equation in standard form: ax² + bx + c = 0.
2. Factor the quadratic expression: Find two binomials that multiply together to give the original quadratic equation. This step may require some trial and error.
3. Set each factor equal to zero: Once you have factored the quadratic expression, set each factor equal to zero.
4. Solve for x: Solve each of the resulting equations for x. These will be the solutions to the original quadratic equation.

Examples

Let's look at some examples of how to solve quadratic equations by factoring.

Example 1

Solve the equation x² - 5x + 6 = 0.

1. The equation is already in standard form.
2. We need to find two binomials that multiply together to give x² - 5x + 6. These binomials are (x - 2) and (x - 3).
3. Setting each factor equal to zero, we get:
• x - 2 = 0
• x - 3 = 0
4. Solving for x, we get:
• x = 2
• x = 3

Therefore, the solutions to the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

Example 2

Solve the equation 2x² + 5x - 3 = 0.

1. The equation is already in standard form.
2. We need to find two binomials that multiply together to give 2x² + 5x - 3. These binomials are (2x - 1) and (x + 3).
3. Setting each factor equal to zero, we get:
• 2x - 1 = 0
• x + 3 = 0
4. Solving for x, we get:
• x = 1/2
• x = -3

Therefore, the solutions to the equation 2x² + 5x - 3 = 0 are x = 1/2 and x = -3.

Conclusion

Factoring is a powerful tool for solving quadratic equations. By following the steps outlined above, you can factor any quadratic equation and find its solutions. This method is particularly useful for equations that can be easily factored. Remember to always check your solutions by plugging them back into the original equation to ensure they are correct.