Solving 80x² + 92x - 84 = 0 A Step By Step Guide

by James Vasile 49 views

Hey everyone! Let's dive into solving this quadratic equation. It might look a bit intimidating at first glance with those big numbers, but don't worry, we'll break it down step-by-step. Our mission is to find the values of x that make the equation 80x² + 92x - 84 = 0 true. We'll explore different methods to tackle this, ensuring you grasp the concepts thoroughly. Quadratic equations are fundamental in mathematics and have real-world applications in physics, engineering, and even economics, so mastering them is super beneficial. So, let's get started and unlock the solutions together!

Understanding Quadratic Equations

First things first, let's make sure we're all on the same page about what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This basically means that the highest power of the variable (in our case, x) is 2. 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 0. If a were 0, the equation would become a linear equation, not a quadratic. In our given equation, 80x² + 92x - 84 = 0, we can identify a = 80, b = 92, and c = -84. These coefficients play a crucial role in determining the solutions to the equation.

The solutions to a quadratic equation are also known as its roots or zeros. These are the values of x that satisfy the equation, meaning when you plug them back into the equation, the left side equals zero. A quadratic equation can have two distinct real roots, one repeated real root, or two complex roots. The nature of the roots depends on the discriminant, which we'll touch upon later. Understanding the basic structure and terminology of quadratic equations is essential before we jump into solving them. Knowing what we're dealing with makes the solution process much smoother and less confusing. Now that we have a solid understanding of the basics, let’s move on to the different methods we can use to solve our equation.

Methods to Solve Quadratic Equations

There are several methods we can use to solve quadratic equations, and each has its own advantages and situations where it's most effective. The three primary methods are:

  1. Factoring: This method involves expressing the quadratic expression 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's applicable, but it's not always straightforward, especially when the coefficients are large or the roots are not rational numbers.

  2. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side. It's a bit more involved than factoring, but it's a powerful technique that always works, regardless of whether the equation can be factored easily. Completing the square is also useful for deriving the quadratic formula, which is another method we'll discuss.

  3. Quadratic Formula: The quadratic formula is a general formula that provides the solutions to any quadratic equation. It's derived from the method of completing the square and is a reliable way to find the roots, even when factoring is difficult or impossible. The formula is x = (-b ± √(b² - 4ac)) / (2a). This formula is a must-know for anyone dealing with quadratic equations.

For our equation, 80x² + 92x - 84 = 0, we'll explore factoring first, as it's often the simplest method if it works. If factoring proves too challenging, we'll move on to the quadratic formula, which will always give us the solutions. Each method provides a unique perspective on solving quadratic equations, so understanding all three will make you a more versatile problem solver.

Attempting to Solve by Factoring

Let's try to solve the equation 80x² + 92x - 84 = 0 by factoring. Factoring involves breaking down the quadratic expression into two binomials that, when multiplied together, give us the original equation. The first step in factoring is often to look for a common factor among the coefficients. In our case, 80, 92, and -84 all share a common factor of 4. Dividing the entire equation by 4 simplifies the equation and makes it easier to work with:

(80x² + 92x - 84) / 4 = 0 / 4

This simplifies to:

20x² + 23x - 21 = 0

Now, we need to find two binomials of the form (Ax + B)(Cx + D) such that their product equals 20x² + 23x - 21. This involves finding two numbers that multiply to give 20 (the coefficient of x²) and two numbers that multiply to give -21 (the constant term). We also need to ensure that the sum of the outer and inner products of the binomials equals 23 (the coefficient of x).

This can be a bit of a trial-and-error process. We need to consider the factors of 20 (1, 2, 4, 5, 10, 20) and the factors of -21 (-1, 1, -3, 3, -7, 7, -21, 21). We're looking for a combination that works when we expand the binomials. After some careful consideration, we can see that the factors (4x + 7) and (5x - 3) might work. Let's multiply them out to check:

(4x + 7)(5x - 3) = 20x² - 12x + 35x - 21 = 20x² + 23x - 21

This matches our simplified quadratic equation! So, we've successfully factored the equation.

Now that we have the factored form, we can find the solutions by setting each factor equal to zero. This is based on the principle that if the product of two factors is zero, then at least one of the factors must be zero. This step is crucial in finding the roots of the equation, as it directly leads us to the values of x that make the equation true. Factoring, when successful, provides a straightforward path to the solutions, making it a valuable technique in our problem-solving toolkit.

Finding the Solutions from Factors

Now that we have the factored form of the equation, (4x + 7)(5x - 3) = 0, we can find the solutions by setting each factor equal to zero. This is because if the product of two factors is zero, then at least one of the factors must be zero.

First, let's set the first factor equal to zero:

4x + 7 = 0

To solve for x, we subtract 7 from both sides:

4x = -7

Then, we divide both sides by 4:

x = -7/4

So, one solution is x = -7/4. Now, let's set the second factor equal to zero:

5x - 3 = 0

To solve for x, we add 3 to both sides:

5x = 3

Then, we divide both sides by 5:

x = 3/5

So, the second solution is x = 3/5. Therefore, the solutions to the equation 80x² + 92x - 84 = 0 are x = -7/4 and x = 3/5. These values of x are the roots of the quadratic equation, meaning they are the points where the parabola represented by the equation intersects the x-axis. Finding these solutions is the ultimate goal when solving quadratic equations, and by setting each factor to zero, we've successfully determined the values of x that satisfy the equation. This step is a direct application of the zero-product property, a fundamental concept in algebra that allows us to solve equations in factored form. With our solutions in hand, we've completed the factoring method and found the roots of the quadratic equation.

Verifying the Solutions

It's always a good practice to verify our solutions to make sure they are correct. To do this, we substitute each value of x back into the original equation and check if it holds true. Let's start with x = -7/4:

80(-7/4)² + 92(-7/4) - 84 = 0

First, we square -7/4:

(-7/4)² = 49/16

Now, substitute this back into the equation:

80(49/16) + 92(-7/4) - 84 = 0

Multiply:

(80 * 49) / 16 - (92 * 7) / 4 - 84 = 0

Simplify:

245 - 161 - 84 = 0

245 - 245 = 0

0 = 0

So, x = -7/4 is indeed a solution. Now, let's verify x = 3/5:

80(3/5)² + 92(3/5) - 84 = 0

First, we square 3/5:

(3/5)² = 9/25

Now, substitute this back into the equation:

80(9/25) + 92(3/5) - 84 = 0

Multiply:

(80 * 9) / 25 + (92 * 3) / 5 - 84 = 0

Simplify:

28.8 + 55.2 - 84 = 0

84 - 84 = 0

0 = 0

So, x = 3/5 is also a solution. By verifying our solutions, we can be confident that we have correctly solved the quadratic equation. This step is not just a formality; it's a crucial part of the problem-solving process that ensures accuracy and reinforces our understanding of the equation. Substituting the solutions back into the original equation and confirming that the equation holds true gives us a sense of completion and validates our efforts.

Conclusion

Alright, guys! We've successfully solved the quadratic equation 80x² + 92x - 84 = 0. We started by understanding the basics of quadratic equations and the different methods we can use to solve them. We then tackled the equation by factoring, which involved simplifying the equation, finding the right factors, and setting each factor to zero to find the solutions. Finally, we verified our solutions to ensure they were correct. The solutions we found were x = -7/4 and x = 3/5. Solving quadratic equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts and real-world applications. Remember, practice makes perfect, so keep solving those equations!