Solving (5x+4)(x+1)=2 A Step-by-Step Guide

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In this article, we will delve into the process of solving the equation (5x + 4)(x + 1) = 2 using a method of your choice. This equation is a quadratic equation, and we will explore how to solve it effectively. Quadratic equations are fundamental in mathematics and have numerous applications in various fields, including physics, engineering, and economics. Understanding how to solve them is a crucial skill for anyone studying these disciplines. Before we proceed, let's understand the fundamental concepts behind solving quadratic equations.

Understanding Quadratic Equations

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 a is not equal to zero. The solutions to a quadratic equation are also known as its roots or zeros. These roots are the values of x that satisfy the equation. In other words, when you substitute a root back into the equation, it makes the equation true. Finding these roots is the primary goal of solving a quadratic equation.

There are several methods to solve quadratic equations, each with its own advantages and disadvantages. Some of the most common methods include:

  • Factoring: This method involves expressing the quadratic equation as a product of two linear factors. If we can factor the equation, we can easily find the roots by setting each factor equal to zero and solving for x. Factoring is often the quickest method when it is applicable, but it is not always possible to factor every quadratic equation.
  • Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial on one side of the equation. Once we have a perfect square, we can take the square root of both sides and solve for x. Completing the square is a reliable method that always works, but it can be more time-consuming than factoring.
  • Quadratic Formula: The quadratic formula is a general formula that can be used to solve any quadratic equation. It provides a direct way to find the roots using the coefficients a, b, and c from the general form of the equation. The quadratic formula is a powerful tool, but it requires careful application to avoid errors.

In our case, the equation (5x + 4)(x + 1) = 2 is not in the standard form ax² + bx + c = 0. Therefore, our first step will be to expand and simplify the equation to bring it into the standard form. This will allow us to identify the coefficients a, b, and c, which we will need to apply the various solution methods.

Expanding and Simplifying the Equation

Our initial equation is (5x + 4)(x + 1) = 2. To put this equation in standard form, we first need to expand the left side of the equation by multiplying the two binomials. We can use the distributive property (also known as the FOIL method) to do this. The FOIL method stands for First, Outer, Inner, Last, and it helps us remember to multiply each term in the first binomial by each term in the second binomial:

  • First: Multiply the first terms of each binomial: 5x * x = 5x²
  • Outer: Multiply the outer terms of the binomials: 5x * 1 = 5x
  • Inner: Multiply the inner terms of the binomials: 4 * x = 4x
  • Last: Multiply the last terms of each binomial: 4 * 1 = 4

Adding these products together gives us: 5x² + 5x + 4x + 4. Now we can simplify this expression by combining like terms. The like terms here are 5x and 4x, which add up to 9x. So, our expanded equation is 5x² + 9x + 4 = 2. This is a crucial step in solving the equation, as it transforms the equation into a form that is more manageable.

Next, we need to move the constant term on the right side of the equation to the left side to set the equation equal to zero. To do this, we subtract 2 from both sides of the equation: 5x² + 9x + 4 - 2 = 2 - 2. This simplifies to 5x² + 9x + 2 = 0. Now we have the quadratic equation in the standard form ax² + bx + c = 0, where a = 5, b = 9, and c = 2. With the equation in this form, we can proceed to solve it using one of the methods we discussed earlier.

Solving the Equation by Factoring

Now that we have the equation in the standard form 5x² + 9x + 2 = 0, let's attempt to solve it by factoring. Factoring involves finding two binomials that, when multiplied together, give us the quadratic expression. This method is often the quickest way to solve a quadratic equation if it can be factored easily. To factor the quadratic expression 5x² + 9x + 2, we need to find two numbers that multiply to give the product of the leading coefficient (5) and the constant term (2), which is 10, and add up to the middle coefficient (9). These two numbers are 1 and 10.

We can rewrite the middle term (9x) as the sum of 1x and 10x: 5x² + 1x + 10x + 2 = 0. Now we can use a technique called factoring by grouping. We group the first two terms and the last two terms together: (5x² + 1x) + (10x + 2) = 0. Next, we factor out the greatest common factor (GCF) from each group.

From the first group, (5x² + 1x), the GCF is x, so we factor out x: x(5x + 1). From the second group, (10x + 2), the GCF is 2, so we factor out 2: 2(5x + 1). Now our equation looks like this: x(5x + 1) + 2(5x + 1) = 0. Notice that both terms have a common factor of (5x + 1). We can factor out this common factor to get: (5x + 1)(x + 2) = 0.

We have now successfully factored the quadratic equation. To find the solutions, we set each factor equal to zero and solve for x:

  1. 5x + 1 = 0 Subtract 1 from both sides: 5x = -1 Divide by 5: x = -1/5
  2. x + 2 = 0 Subtract 2 from both sides: x = -2

Therefore, the solutions to the equation (5x + 4)(x + 1) = 2 are x = -1/5 and x = -2. Factoring allowed us to find these solutions in a straightforward manner, demonstrating the effectiveness of this method when applicable.

Solving the Equation Using the Quadratic Formula

While we successfully solved the equation 5x² + 9x + 2 = 0 by factoring, it's also beneficial to demonstrate how to solve it using the quadratic formula. The quadratic formula is a versatile tool that can be used to solve any quadratic equation, regardless of whether it can be factored easily. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. In our equation, 5x² + 9x + 2 = 0, we have a = 5, b = 9, and c = 2. To apply the quadratic formula, we substitute these values into the formula:

x = (-9 ± √(9² - 4 * 5 * 2)) / (2 * 5)

First, we simplify the expression under the square root:

9² - 4 * 5 * 2 = 81 - 40 = 41

So, our equation becomes:

x = (-9 ± √41) / 10

Now, we can find the two solutions by considering both the plus and minus signs:

  1. x = (-9 + √41) / 10 √41 is approximately 6.403. So, x ≈ (-9 + 6.403) / 10 = -2.597 / 10 ≈ -0.2597 ≈ -1/5
  2. x = (-9 - √41) / 10 x ≈ (-9 - 6.403) / 10 = -15.403 / 10 ≈ -1.5403 ≈ -2

As you can see, the solutions obtained using the quadratic formula are approximately x = -1/5 and x = -2, which match the solutions we found by factoring. The quadratic formula confirms our earlier results and demonstrates its reliability in solving quadratic equations. This method is particularly useful when dealing with equations that are difficult or impossible to factor.

Conclusion

In this article, we have thoroughly explored how to solve the equation (5x + 4)(x + 1) = 2. We began by expanding and simplifying the equation to bring it into the standard quadratic form 5x² + 9x + 2 = 0. Then, we solved the equation using two different methods: factoring and the quadratic formula. Both methods yielded the same solutions, x = -1/5 and x = -2, demonstrating the versatility and effectiveness of these techniques.

Understanding how to solve quadratic equations is a fundamental skill in mathematics with wide-ranging applications. Whether you choose to use factoring, completing the square, or the quadratic formula, the key is to apply the method correctly and carefully. By mastering these techniques, you will be well-equipped to tackle more complex mathematical problems in various fields. The ability to solve quadratic equations not only strengthens your mathematical foundation but also enhances your problem-solving skills in general.

Quadratic equations appear in various real-world scenarios, such as projectile motion, optimization problems, and circuit analysis. Therefore, having a solid understanding of how to solve them is essential for anyone pursuing a career in science, engineering, or mathematics. We encourage you to practice solving quadratic equations using different methods to develop your proficiency and confidence in this area.

By mastering the techniques discussed in this article, you will be well-prepared to solve a wide range of quadratic equations and apply these skills to real-world problems. Remember, practice is key to success in mathematics, so continue to challenge yourself with new problems and explore different approaches to find the solutions. With consistent effort and a clear understanding of the underlying concepts, you can confidently tackle any quadratic equation that comes your way.