Solving Quadratic Equations Step By Step 5x² - 13x - 6 = 0
Solving quadratic equations is a fundamental skill in algebra. In this article, we will walk through a step-by-step solution to the quadratic equation 5x² - 13x - 6 = 0. This method involves factoring the quadratic expression, which is a common and effective technique for finding the roots of a quadratic equation. Understanding this process is crucial for anyone studying algebra or related fields. The steps provided break down the solution into manageable parts, making it easier to follow and understand. Let's dive into how we can solve this equation and explore the underlying principles of quadratic equations.
Understanding Quadratic Equations
Before we jump into the solution, let's briefly discuss what a quadratic equation is and why it's important. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we are solving for. Quadratic equations appear in various areas of mathematics and physics, making their solutions essential in many applications. From calculating trajectories to designing structures, understanding quadratic equations is invaluable.
The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the x-values where the parabola described by the quadratic equation intersects the x-axis. A quadratic equation can have two, one, or no real roots, depending on the discriminant (b² - 4ac). If the discriminant is positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are no real roots (but there are two complex roots). The method we will use in this article, factoring, is one of several ways to find these roots. Other methods include the quadratic formula and completing the square. Each method has its strengths and weaknesses, but factoring is often the quickest and most straightforward when the quadratic expression can be factored easily.
Factoring involves breaking down the quadratic expression into the product of two binomials. This method relies on reversing the process of expanding two binomials using the distributive property (also known as FOIL - First, Outer, Inner, Last). When factoring, we look for two numbers that multiply to the product of the leading coefficient (a) and the constant term (c), and that add up to the middle coefficient (b). Finding these numbers is the key step in the factoring process, and it's what we'll focus on in the solution below. By mastering factoring, you'll be well-equipped to solve a wide range of quadratic equations efficiently.
Step 1: Product of Coefficients
The first step in solving the quadratic equation 5x² - 13x - 6 = 0 by factoring is to find the product of the coefficients of the x² term and the constant term. In this equation, the coefficient of the x² term is 5, and the constant term is -6. Multiplying these two coefficients gives us:
5 × (-6) = -30
This product, -30, is a crucial value because it helps us identify the two numbers needed to split the middle term. The middle term is the term with x, which in this case is -13x. We are looking for two numbers that not only multiply to -30 but also add up to -13. This is a classic problem in factoring quadratic equations and requires a bit of number sense and trial and error.
The significance of this step lies in the underlying principle of factoring quadratic expressions. When we factor a quadratic equation in the form ax² + bx + c = 0, we are essentially trying to reverse the process of expanding two binomials. The product of the coefficients, ac, gives us a target product for the constant terms in the binomials, while the coefficient of the middle term, b, gives us a target sum. By finding two numbers that satisfy both conditions, we can rewrite the middle term in a way that allows us to factor by grouping. This method is particularly effective when the coefficients are integers and the quadratic expression can be factored neatly.
Finding the correct pair of numbers may require some practice and familiarity with factors. A systematic approach can be helpful. Start by listing the pairs of factors of the product (-30) and then check which pair adds up to the middle coefficient (-13). This process might involve trying different combinations until the right pair is found. For instance, you might consider pairs like (1, -30), (-1, 30), (2, -15), (-2, 15), and so on. By carefully examining these pairs, you can identify the one that meets both criteria: multiplying to -30 and adding to -13. This initial step sets the stage for the rest of the factoring process and is essential for finding the roots of the quadratic equation.
Step 2: Split Middle Term
Now that we have the product of the coefficients (-30), we need to find two numbers that multiply to -30 and add up to the coefficient of the middle term, which is -13. Through trial and error or by listing factors, we find that the numbers -15 and 2 satisfy these conditions:
-15 × 2 = -30 -15 + 2 = -13
These two numbers, -15 and 2, are the key to splitting the middle term. Instead of writing -13x, we will rewrite it as -15x + 2x. This allows us to rewrite the original equation as:
5x² - 15x + 2x - 6 = 0
Splitting the middle term is a crucial step in the factoring process because it sets up the equation for factoring by grouping. By breaking the middle term into two parts, we create four terms in the equation, which can then be grouped into pairs. Each pair can be factored separately, and if done correctly, the factored pairs will share a common binomial factor. This common binomial factor can then be factored out, leading to the factored form of the quadratic expression.
The reason this method works lies in the distributive property. When we split the middle term using the correct numbers, we are essentially undoing the distributive property that was used when the two binomials were multiplied to form the quadratic expression. By rewriting the middle term, we create a structure that allows us to reverse this process and identify the original binomial factors. This step requires careful attention to the signs and magnitudes of the numbers involved. A mistake in this step can lead to an incorrect factorization and, consequently, incorrect roots for the quadratic equation.
Furthermore, the choice of numbers used to split the middle term is not arbitrary. They must satisfy both the multiplication and addition conditions described earlier. This ensures that the rewritten equation is equivalent to the original equation and that the factoring process will lead to the correct solution. Understanding this principle is essential for mastering the factoring technique and applying it to various quadratic equations. The ability to split the middle term effectively is a fundamental skill in algebra and is crucial for solving quadratic equations by factoring.
Step 3: Rewrite Equation
In the previous step, we identified the two numbers, -15 and 2, that allow us to split the middle term of the quadratic equation. Now, we rewrite the equation 5x² - 13x - 6 = 0 using these numbers to replace the middle term -13x. This gives us:
5x² - 15x + 2x - 6 = 0
This rewritten equation is equivalent to the original equation, but it is now structured in a way that allows us to factor by grouping. The act of rewriting the equation might seem like a small step, but it is a pivotal moment in the factoring process. It transforms the equation from a simple quadratic form into a format that is amenable to further manipulation. The key here is to ensure that the rewritten equation maintains the same value as the original equation. This is achieved by carefully replacing the middle term with its equivalent expression, which we found by splitting the middle term using the correct numbers.
The significance of this step is that it sets the stage for the next step, which is grouping the terms. By having four terms instead of three, we can group them into pairs and factor out common factors from each pair. This process reveals the common binomial factor that is essential for completing the factorization. Without this rewriting step, factoring by grouping would not be possible.
Furthermore, this step highlights the importance of understanding the properties of algebraic expressions. The ability to manipulate expressions while preserving their value is a fundamental skill in algebra. Rewriting the equation by splitting the middle term is a practical application of this skill. It demonstrates how an expression can be transformed into a different form to facilitate further operations. This step also underscores the importance of accuracy and attention to detail. Any mistake in rewriting the equation can lead to an incorrect factorization and, ultimately, incorrect solutions. Therefore, it is crucial to double-check that the rewritten equation is indeed equivalent to the original equation before proceeding with the next step.
Step 4: Group Terms
After rewriting the equation as 5x² - 15x + 2x - 6 = 0, the next step is to group the terms. This involves pairing the first two terms and the last two terms together. We use parentheses to indicate these groups:
(5x² - 15x) + (2x - 6) = 0
Grouping terms is a strategic move that prepares the equation for the next phase of factoring, which is factoring out the greatest common factor (GCF) from each group. The purpose of grouping is to isolate common factors within pairs of terms, making it easier to identify and extract those factors. This technique is a cornerstone of factoring by grouping and is essential for solving quadratic equations that can be factored in this way.
The choice of how to group terms is not arbitrary. We group the terms in a way that allows us to factor out a common factor from each group. In this case, the first group (5x² - 15x) shares a common factor of 5x, and the second group (2x - 6) shares a common factor of 2. By grouping these terms together, we set ourselves up to simplify the equation further.
The parentheses in the grouped equation serve an important purpose. They indicate that the terms within each group should be treated as a single unit. This helps to maintain the correct order of operations and ensures that we apply the distributive property correctly when factoring out the GCF. The plus sign between the groups indicates that we are adding the results of factoring each group, which is consistent with the original equation.
This step may seem straightforward, but it is a critical bridge between splitting the middle term and factoring out the GCF. Grouping terms correctly sets the stage for the subsequent factoring steps and is a key component of solving quadratic equations by factoring. By mastering this technique, you can effectively simplify complex quadratic expressions and pave the way for finding their roots.
Step 5: Factor
Now that we have grouped the terms as (5x² - 15x) + (2x - 6) = 0, the next step is to factor out the greatest common factor (GCF) from each group. Let's start with the first group, (5x² - 15x). The GCF of 5x² and -15x is 5x. Factoring out 5x, we get:
5x(x - 3)
Next, we factor out the GCF from the second group, (2x - 6). The GCF of 2x and -6 is 2. Factoring out 2, we get:
2(x - 3)
So, the equation becomes:
5x(x - 3) + 2(x - 3) = 0
Factoring out the GCF from each group is a critical step in the factoring process. It allows us to simplify each group and, more importantly, reveals a common binomial factor. In this case, the common binomial factor is (x - 3). This common factor is the key to completing the factorization of the quadratic expression.
The process of finding the GCF involves identifying the largest factor that divides evenly into all terms within the group. This requires understanding the factors of both the coefficients and the variables. For example, in the group (5x² - 15x), the largest factor that divides both 5x² and -15x is 5x. Similarly, in the group (2x - 6), the largest factor that divides both 2x and -6 is 2.
By factoring out the GCF, we are essentially reversing the distributive property. We are taking an expression that has been expanded and breaking it down into its constituent factors. This process is fundamental to factoring and is essential for solving quadratic equations and other algebraic problems. The success of this step depends on accurately identifying the GCF and applying the distributive property in reverse.
Furthermore, this step highlights the importance of pattern recognition in mathematics. By recognizing the common binomial factor, we can see the structure of the factored expression and proceed with the final steps of the solution. The ability to factor effectively is a valuable skill in algebra and is crucial for simplifying expressions, solving equations, and tackling more advanced mathematical concepts. In this instance, this step paves the way for the final factoring and solution of the quadratic equation.
