Solving Quadratic Equations by Factoring

Quadratic equations are equations that contain a variable squared. They are often written in the form ax² + bx + c = 0, where a, b, and c are constants. Solving quadratic equations can be done using a variety of methods, including factoring, the quadratic formula, and completing the square. In this blog post, we will focus on solving quadratic equations by factoring.

What is Factoring?

Factoring is the process of breaking down a polynomial into its factors. Factors are expressions that multiply together to give the original polynomial. For example, the factors of the polynomial x² + 5x + 6 are (x + 2) and (x + 3), because (x + 2)(x + 3) = x² + 5x + 6.

How to Solve Quadratic Equations by Factoring

To solve a quadratic equation by factoring, follow these steps:

1. Write the quadratic equation in standard form. This means that the equation should be in the form ax² + bx + c = 0.
2. Factor the quadratic expression. This means finding two expressions that multiply together to give the original quadratic expression.
3. Set each factor equal to zero. This will give you two linear equations.
4. Solve each linear equation for x. This will give you the two solutions to the quadratic equation.

Example

Let's solve the quadratic equation x² + 5x + 6 = 0 by factoring.

1. Write the quadratic equation in standard form. The equation is already in standard form.
2. Factor the quadratic expression. The factors of x² + 5x + 6 are (x + 2) and (x + 3).
3. Set each factor equal to zero. This gives us the equations x + 2 = 0 and x + 3 = 0.
4. Solve each linear equation for x. Solving x + 2 = 0 gives us x = -2. Solving x + 3 = 0 gives us x = -3.

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

Tips for Factoring Quadratic Expressions

Here are some tips for factoring quadratic expressions:

• Look for common factors. If the terms in the quadratic expression have a common factor, factor it out first.
• Use the FOIL method. The FOIL method is a mnemonic device that helps you factor quadratic expressions. FOIL stands for First, Outer, Inner, Last. To use the FOIL method, multiply the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms.
• Use the difference of squares pattern. The difference of squares pattern is a special factoring pattern that can be used to factor quadratic expressions that are in the form a² - b². The pattern is a² - b² = (a + b)(a - b).
• Use the perfect square trinomial pattern. The perfect square trinomial pattern is another special factoring pattern that can be used to factor quadratic expressions that are in the form a² + 2ab + b² or a² - 2ab + b². The patterns are a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)².

Conclusion

Solving quadratic equations by factoring is a valuable skill that can be used to solve a variety of problems. By following the steps outlined in this blog post, you can learn how to factor quadratic expressions and solve quadratic equations using this method.