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

In mathematics, quadratic equations are polynomial equations of the second degree. They are of the general form ax² + bx + c = 0, where a, b, and c are constants, and x represents an unknown variable. Solving quadratic equations involves finding the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. One of the fundamental methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two linear factors. This method is particularly effective when the quadratic equation can be factored easily. In this article, we will demonstrate how to solve various quadratic equations by factoring, providing step-by-step explanations to enhance understanding and mastery of this technique. The ability to solve quadratic equations is crucial in various fields of science, engineering, and mathematics, making it a fundamental skill for students and professionals alike.

1. Solving x² + 9x - 36 = 0 by Factoring

To solve the quadratic equation x² + 9x - 36 = 0 by factoring, the first crucial step involves identifying two numbers that multiply to give the constant term (-36) and add up to the coefficient of the linear term (9). This is a critical part of the factoring process, as it sets the stage for expressing the quadratic equation as a product of two binomials. By carefully considering the factors of -36, we can determine the pair that satisfies both conditions. In this case, the numbers 12 and -3 fit the criteria perfectly, since 12 multiplied by -3 equals -36, and 12 plus -3 equals 9. Once we have identified these numbers, the next step is to rewrite the middle term (9x) of the quadratic equation using these numbers. So, we rewrite 9x as 12x - 3x, transforming the equation x² + 9x - 36 = 0 into x² + 12x - 3x - 36 = 0. This transformation allows us to group terms and factor by grouping, which is a standard technique in solving quadratic equations.

After rewriting the equation, the next step is to factor by grouping. We group the first two terms and the last two terms together: (x² + 12x) + (-3x - 36) = 0. From the first group, (x² + 12x), we can factor out x, and from the second group, (-3x - 36), we can factor out -3. This gives us x(x + 12) - 3(x + 12) = 0. Now, we observe that (x + 12) is a common factor in both terms. Factoring out the common factor (x + 12), we get (x + 12)(x - 3) = 0. This factored form of the quadratic equation is a significant achievement, as it directly leads to identifying the solutions. The equation is now expressed as a product of two binomials, making it easy to apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is the key to finding the values of x that satisfy the original quadratic equation.

Now that we have the factored form (x + 12)(x - 3) = 0, we apply the zero-product property. This means setting each factor equal to zero and solving for x. First, we set (x + 12) = 0, which gives us x = -12. Second, we set (x - 3) = 0, which gives us x = 3. These values, x = -12 and x = 3, are the solutions to the quadratic equation x² + 9x - 36 = 0. To verify these solutions, we can substitute each value back into the original equation and check if the equation holds true. Substituting x = -12 into the equation gives us (-12)² + 9(-12) - 36 = 144 - 108 - 36 = 0, which confirms that x = -12 is a solution. Similarly, substituting x = 3 into the equation gives us (3)² + 9(3) - 36 = 9 + 27 - 36 = 0, confirming that x = 3 is also a solution. Therefore, the solutions to the quadratic equation x² + 9x - 36 = 0 are x = -12 and x = 3. This step-by-step factoring method provides a clear and effective way to solve quadratic equations, demonstrating the power of algebraic manipulation in finding solutions.

2. Solving x² + 6x - 27 = 0 by Factoring

To solve the quadratic equation x² + 6x - 27 = 0 by factoring, we follow a similar process as before. The first step is to identify two numbers that multiply to -27 (the constant term) and add up to 6 (the coefficient of the x term). This is a critical step in factoring quadratic equations, as the correct identification of these numbers will allow us to rewrite the middle term and proceed with factoring by grouping. By systematically considering the factors of -27, we can find the pair that meets both criteria. In this case, the numbers 9 and -3 are the appropriate choice because 9 multiplied by -3 is -27, and 9 plus -3 is 6. Once these numbers are identified, we can proceed to rewrite the middle term of the quadratic equation using these numbers.

Once we have identified the numbers 9 and -3, the next step is to rewrite the middle term (6x) of the quadratic equation using these numbers. This means we replace 6x with 9x - 3x, which transforms the original equation x² + 6x - 27 = 0 into x² + 9x - 3x - 27 = 0. This transformation is crucial because it allows us to factor the equation by grouping, a common and effective method for solving quadratic equations. By rewriting the equation in this way, we set the stage for grouping the terms and factoring out common factors, which will ultimately lead us to the solutions of the equation. Factoring by grouping is a powerful technique that simplifies the process of solving quadratic equations, especially when the equation can be easily factored.

With the equation rewritten as x² + 9x - 3x - 27 = 0, we now proceed to factor by grouping. We group the first two terms and the last two terms: (x² + 9x) + (-3x - 27) = 0. From the first group, (x² + 9x), we can factor out x, resulting in x(x + 9). From the second group, (-3x - 27), we can factor out -3, resulting in -3(x + 9). The equation now becomes x(x + 9) - 3(x + 9) = 0. Notice that (x + 9) is a common factor in both terms. Factoring out the common factor (x + 9), we get (x + 9)(x - 3) = 0. This factored form of the quadratic equation is a critical step towards finding the solutions. It simplifies the equation into a product of two binomials, which allows us to easily apply the zero-product property. This property is the key to determining the values of x that satisfy the original equation.

Now that we have the factored form (x + 9)(x - 3) = 0, we apply the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x. First, we set (x + 9) = 0, which gives us x = -9. Second, we set (x - 3) = 0, which gives us x = 3. These values, x = -9 and x = 3, are the solutions to the quadratic equation x² + 6x - 27 = 0. To verify these solutions, we can substitute each value back into the original equation and check if the equation holds true. Substituting x = -9 into the equation gives us (-9)² + 6(-9) - 27 = 81 - 54 - 27 = 0, which confirms that x = -9 is a solution. Similarly, substituting x = 3 into the equation gives us (3)² + 6(3) - 27 = 9 + 18 - 27 = 0, confirming that x = 3 is also a solution. Therefore, the solutions to the quadratic equation x² + 6x - 27 = 0 are x = -9 and x = 3. This methodical approach to factoring quadratic equations ensures accuracy and provides a clear understanding of the solution process.

3. Solving x² + 4x - 60 = 0 by Factoring

To solve the quadratic equation x² + 4x - 60 = 0 by factoring, the initial step involves identifying two numbers that multiply to -60 (the constant term) and add up to 4 (the coefficient of the linear term). This is a foundational step in the factoring process, as the correct identification of these numbers will enable us to rewrite the middle term and factor the equation by grouping. By meticulously examining the factors of -60, we can pinpoint the pair that satisfies both conditions. In this case, the numbers 10 and -6 are the appropriate choice because 10 multiplied by -6 is -60, and 10 plus -6 is 4. Once we have determined these numbers, we can proceed to rewrite the middle term of the quadratic equation using these numbers. This step is critical for setting up the equation for factoring by grouping.

Having identified the numbers 10 and -6, we now rewrite the middle term (4x) of the quadratic equation using these numbers. This entails replacing 4x with 10x - 6x, thereby transforming the original equation x² + 4x - 60 = 0 into x² + 10x - 6x - 60 = 0. This transformation is a pivotal step as it prepares the equation for factoring by grouping, a standard and highly effective technique for solving quadratic equations. By rewriting the equation in this manner, we pave the way for grouping the terms and extracting common factors, ultimately leading us to the solutions of the equation. Factoring by grouping is a valuable tool that simplifies the process of solving quadratic equations, particularly when the equation is easily factorable. This approach allows us to break down the quadratic equation into manageable parts, making the solution process more straightforward.

With the equation rewritten as x² + 10x - 6x - 60 = 0, we proceed to factor by grouping. We group the first two terms and the last two terms: (x² + 10x) + (-6x - 60) = 0. From the first group, (x² + 10x), we can factor out x, which yields x(x + 10). From the second group, (-6x - 60), we can factor out -6, resulting in -6(x + 10). The equation now becomes x(x + 10) - 6(x + 10) = 0. We observe that (x + 10) is a common factor in both terms. Factoring out the common factor (x + 10), we obtain (x + 10)(x - 6) = 0. This factored form of the quadratic equation is a crucial milestone in finding the solutions. It simplifies the equation into a product of two binomials, allowing us to easily apply the zero-product property, which is essential for determining the values of x that satisfy the original equation.

Now that we have the factored form (x + 10)(x - 6) = 0, we apply the zero-product property. According to this property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x. First, we set (x + 10) = 0, which gives us x = -10. Second, we set (x - 6) = 0, which gives us x = 6. These values, x = -10 and x = 6, are the solutions to the quadratic equation x² + 4x - 60 = 0. To verify these solutions, we can substitute each value back into the original equation and check if the equation holds true. Substituting x = -10 into the equation gives us (-10)² + 4(-10) - 60 = 100 - 40 - 60 = 0, which confirms that x = -10 is a solution. Similarly, substituting x = 6 into the equation gives us (6)² + 4(6) - 60 = 36 + 24 - 60 = 0, confirming that x = 6 is also a solution. Thus, the solutions to the quadratic equation x² + 4x - 60 = 0 are x = -10 and x = 6. This systematic approach to factoring quadratic equations ensures a clear and accurate solution process.

4. Solving 2x² - 5x - 18 = 0 by Factoring

To solve the quadratic equation 2x² - 5x - 18 = 0 by factoring, the process is slightly different from the previous examples because the coefficient of the x² term is not 1. The initial step involves multiplying the coefficient of the x² term (2) by the constant term (-18), which gives us -36. We then need to identify two numbers that multiply to -36 and add up to -5 (the coefficient of the x term). This step is crucial for factoring quadratic equations where the leading coefficient is not 1, as it helps in rewriting the middle term appropriately. By systematically examining the factors of -36, we can find the pair that meets both criteria. In this case, the numbers -9 and 4 are the appropriate choice because -9 multiplied by 4 is -36, and -9 plus 4 is -5. Once these numbers are identified, we can proceed to rewrite the middle term of the quadratic equation using these numbers.

After identifying the numbers -9 and 4, the next step is to rewrite the middle term (-5x) of the quadratic equation using these numbers. This means we replace -5x with -9x + 4x, transforming the original equation 2x² - 5x - 18 = 0 into 2x² - 9x + 4x - 18 = 0. This transformation is essential for factoring by grouping, a common technique used to solve quadratic equations when the leading coefficient is not 1. By rewriting the equation in this manner, we prepare it for the next step, which involves grouping terms and factoring out common factors. Factoring by grouping allows us to simplify the equation and ultimately find its solutions. This method breaks down the quadratic expression into manageable parts, making the factoring process more straightforward and efficient.

With the equation rewritten as 2x² - 9x + 4x - 18 = 0, we proceed to factor by grouping. We group the first two terms and the last two terms: (2x² - 9x) + (4x - 18) = 0. From the first group, (2x² - 9x), we can factor out x, resulting in x(2x - 9). From the second group, (4x - 18), we can factor out 2, resulting in 2(2x - 9). The equation now becomes x(2x - 9) + 2(2x - 9) = 0. Notice that (2x - 9) is a common factor in both terms. Factoring out the common factor (2x - 9), we get (2x - 9)(x + 2) = 0. This factored form of the quadratic equation is a critical step toward finding the solutions. It simplifies the equation into a product of two binomials, which allows us to easily apply the zero-product property. This property is key to determining the values of x that satisfy the original equation.

Now that we have the factored form (2x - 9)(x + 2) = 0, we apply the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x. First, we set (2x - 9) = 0, which gives us 2x = 9, and then x = 9/2. Second, we set (x + 2) = 0, which gives us x = -2. These values, x = 9/2 and x = -2, are the solutions to the quadratic equation 2x² - 5x - 18 = 0. To verify these solutions, we can substitute each value back into the original equation and check if the equation holds true. Substituting x = 9/2 into the equation gives us 2(9/2)² - 5(9/2) - 18 = 2(81/4) - 45/2 - 18 = 81/2 - 45/2 - 36/2 = 0, which confirms that x = 9/2 is a solution. Similarly, substituting x = -2 into the equation gives us 2(-2)² - 5(-2) - 18 = 2(4) + 10 - 18 = 8 + 10 - 18 = 0, confirming that x = -2 is also a solution. Therefore, the solutions to the quadratic equation 2x² - 5x - 18 = 0 are x = 9/2 and x = -2. This step-by-step approach to factoring quadratic equations with a leading coefficient other than 1 ensures accuracy and provides a clear understanding of the solution process.

5. Solving x² + 6x = 16 by Factoring

To solve the quadratic equation x² + 6x = 16 by factoring, the initial step is to rewrite the equation in the standard form of a quadratic equation, which is ax² + bx + c = 0. This is crucial because factoring methods are typically applied to equations in this standard form. To rewrite the given equation, we subtract 16 from both sides, which transforms the equation x² + 6x = 16 into x² + 6x - 16 = 0. Now that the equation is in standard form, we can proceed with the factoring process. This preparation step ensures that we can effectively apply the techniques of factoring to find the solutions of the quadratic equation. Transforming the equation into standard form is a fundamental step in solving quadratic equations by factoring, as it sets the stage for the subsequent steps.

Now that the equation is in the standard form x² + 6x - 16 = 0, the next step is to identify two numbers that multiply to -16 (the constant term) and add up to 6 (the coefficient of the x term). This is a critical step in the factoring process, as the correct identification of these numbers will allow us to rewrite the middle term and proceed with factoring by grouping. By systematically considering the factors of -16, we can find the pair that meets both criteria. In this case, the numbers 8 and -2 are the appropriate choice because 8 multiplied by -2 is -16, and 8 plus -2 is 6. Once these numbers are identified, we can proceed to rewrite the middle term of the quadratic equation using these numbers. This step is essential for setting up the equation for factoring by grouping, a standard and effective method for solving quadratic equations.

Having identified the numbers 8 and -2, we now rewrite the middle term (6x) of the quadratic equation using these numbers. This means we replace 6x with 8x - 2x, which transforms the equation x² + 6x - 16 = 0 into x² + 8x - 2x - 16 = 0. This transformation is a crucial step as it prepares the equation for factoring by grouping, a common technique for solving quadratic equations. By rewriting the equation in this manner, we set the stage for grouping the terms and factoring out common factors, which will ultimately lead us to the solutions of the equation. Factoring by grouping is a powerful method that simplifies the process of solving quadratic equations, especially when the equation can be easily factored. This approach allows us to break down the quadratic expression into manageable components, making the solution process more straightforward and efficient.

With the equation rewritten as x² + 8x - 2x - 16 = 0, we proceed to factor by grouping. We group the first two terms and the last two terms: (x² + 8x) + (-2x - 16) = 0. From the first group, (x² + 8x), we can factor out x, resulting in x(x + 8). From the second group, (-2x - 16), we can factor out -2, resulting in -2(x + 8). The equation now becomes x(x + 8) - 2(x + 8) = 0. Notice that (x + 8) is a common factor in both terms. Factoring out the common factor (x + 8), we get (x + 8)(x - 2) = 0. This factored form of the quadratic equation is a crucial milestone in finding the solutions. It simplifies the equation into a product of two binomials, allowing us to easily apply the zero-product property, which is essential for determining the values of x that satisfy the original equation.

Now that we have the factored form (x + 8)(x - 2) = 0, we apply the zero-product property. According to this property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x. First, we set (x + 8) = 0, which gives us x = -8. Second, we set (x - 2) = 0, which gives us x = 2. These values, x = -8 and x = 2, are the solutions to the quadratic equation x² + 6x = 16. To verify these solutions, we can substitute each value back into the original equation and check if the equation holds true. Substituting x = -8 into the equation gives us (-8)² + 6(-8) = 64 - 48 = 16, which confirms that x = -8 is a solution. Similarly, substituting x = 2 into the equation gives us (2)² + 6(2) = 4 + 12 = 16, confirming that x = 2 is also a solution. Thus, the solutions to the quadratic equation x² + 6x = 16 are x = -8 and x = 2. This systematic approach to factoring quadratic equations ensures a clear and accurate solution process.

In conclusion, solving quadratic equations by factoring is a fundamental algebraic technique that involves expressing the quadratic expression as a product of two linear factors. This method relies on identifying two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. The process typically involves rewriting the middle term, factoring by grouping, and applying the zero-product property to find the solutions. The examples provided illustrate the step-by-step application of this method to various quadratic equations, demonstrating its effectiveness and versatility. Mastering factoring techniques is essential for solving quadratic equations and forms a crucial foundation for more advanced algebraic concepts. Understanding these methods empowers students and professionals to tackle a wide range of mathematical problems in various fields.