Solving X² = 8 - 5x A Step-by-Step Guide
In the realm of algebra, quadratic equations hold a position of prominence. These equations, characterized by the presence of a variable raised to the power of two, appear in various mathematical and real-world contexts. Understanding how to solve quadratic equations is therefore a fundamental skill for students and professionals alike. This article delves into the process of solving the specific quadratic equation x² = 8 - 5x, providing a step-by-step guide and exploring the underlying principles involved. We will transform the equation into standard form, identify the coefficients, and then apply the quadratic formula to arrive at the solutions. By the end of this exploration, you will have a solid grasp of how to tackle similar quadratic equations with confidence.
Transforming the Equation into Standard Form
The first crucial step in solving any quadratic equation is to rearrange it into its standard form. The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for. Our given equation is x² = 8 - 5x. To transform this into standard form, we need to move all terms to one side of the equation, leaving zero on the other side. This involves adding 5x and subtracting 8 from both sides of the equation. When we perform these operations, we obtain the following:
x² + 5x - 8 = 0
Now, our equation is in the standard form ax² + bx + c = 0. Identifying the coefficients 'a', 'b', and 'c' is the next important step. In our transformed equation, the coefficient of x² (a) is 1, the coefficient of x (b) is 5, and the constant term (c) is -8. These coefficients are crucial for applying the quadratic formula, which is the next stage in our solution process. Correctly identifying 'a', 'b', and 'c' is paramount, as any error here will propagate through the rest of the solution, leading to an incorrect answer. This step sets the foundation for applying the quadratic formula effectively and accurately.
Applying the Quadratic Formula
The quadratic formula is a powerful tool for finding the solutions (also known as roots) of any quadratic equation expressed in the standard form ax² + bx + c = 0. This formula provides a direct method for calculating the values of 'x' that satisfy the equation. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
This formula might seem intimidating at first, but it becomes straightforward when applied systematically. The ± symbol indicates that there are generally two solutions: one obtained by adding the square root term and the other by subtracting it. Now, let's apply this formula to our equation, x² + 5x - 8 = 0. We've already identified the coefficients: a = 1, b = 5, and c = -8. Substituting these values into the quadratic formula, we get:
x = (-5 ± √(5² - 4 * 1 * -8)) / (2 * 1)
Now, we simplify the expression step by step. First, we calculate the term inside the square root:
5² - 4 * 1 * -8 = 25 + 32 = 57
So, our equation becomes:
x = (-5 ± √57) / 2
This gives us two possible solutions for x, which we will explore in the next section.
Calculating the Solutions
Having applied the quadratic formula and simplified the expression, we now have two potential solutions for x: x = (-5 + √57) / 2 and x = (-5 - √57) / 2. These solutions arise from the ± symbol in the quadratic formula, which indicates that a quadratic equation typically has two roots. The square root of 57 is an irrational number, meaning it cannot be expressed as a simple fraction, so we leave it in its radical form for an exact answer. Now, let's write out the two solutions explicitly:
- x₁ = (-5 + √57) / 2
- x₂ = (-5 - √57) / 2
These are the exact solutions to the quadratic equation x² = 8 - 5x. These solutions represent the points where the parabola described by the equation intersects the x-axis. It's important to note that these solutions are irrational numbers due to the presence of the square root of 57. If we needed a decimal approximation, we could use a calculator to find the approximate value of √57 and then complete the calculations. However, leaving the solutions in this form provides the most accurate representation.
Verifying the Solutions
To ensure that the solutions we've calculated are correct, it's a good practice to verify them by substituting them back into the original equation. This process helps catch any potential errors made during the application of the quadratic formula or simplification steps. Let's take our two solutions, x₁ = (-5 + √57) / 2 and x₂ = (-5 - √57) / 2, and substitute them into the original equation, x² = 8 - 5x, one at a time.
Verification of x₁ = (-5 + √57) / 2
Substituting x₁ into the equation, we get:
((-5 + √57) / 2)² = 8 - 5((-5 + √57) / 2)
Let's simplify both sides separately. First, the left side:
((-5 + √57) / 2)² = (25 - 10√57 + 57) / 4 = (82 - 10√57) / 4 = (41 - 5√57) / 2
Now, the right side:
8 - 5((-5 + √57) / 2) = 8 + (25 - 5√57) / 2 = (16 + 25 - 5√57) / 2 = (41 - 5√57) / 2
Since both sides are equal, the solution x₁ is verified.
Verification of x₂ = (-5 - √57) / 2
Substituting x₂ into the equation, we get:
((-5 - √57) / 2)² = 8 - 5((-5 - √57) / 2)
Let's simplify both sides separately. First, the left side:
((-5 - √57) / 2)² = (25 + 10√57 + 57) / 4 = (82 + 10√57) / 4 = (41 + 5√57) / 2
Now, the right side:
8 - 5((-5 - √57) / 2) = 8 + (25 + 5√57) / 2 = (16 + 25 + 5√57) / 2 = (41 + 5√57) / 2
Since both sides are equal, the solution x₂ is also verified.
Both solutions, x₁ = (-5 + √57) / 2 and x₂ = (-5 - √57) / 2, satisfy the original equation. This verification step confirms the accuracy of our calculations and provides confidence in our solutions.
Conclusion
In this comprehensive guide, we have successfully navigated the process of solving the quadratic equation x² = 8 - 5x. We began by transforming the equation into its standard form, x² + 5x - 8 = 0, and identified the coefficients a, b, and c. We then applied the powerful quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, to find the solutions. This led us to the two distinct roots: x₁ = (-5 + √57) / 2 and x₂ = (-5 - √57) / 2. To ensure the accuracy of our results, we meticulously verified both solutions by substituting them back into the original equation, confirming that they indeed satisfy the given condition. Understanding and mastering the techniques for solving quadratic equations is essential in mathematics and its applications. This article has provided a detailed roadmap for tackling such problems, equipping you with the knowledge and confidence to solve similar equations in the future. The quadratic formula is a cornerstone of algebra, and its application extends far beyond textbook exercises, appearing in various scientific and engineering disciplines. By practicing these methods and understanding the underlying principles, you can build a strong foundation in mathematical problem-solving.
