Solving Quadratic Equations: Find The Solutions For X² = 8 - 5x

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Hey guys! Let's dive into solving a quadratic equation today. We're tackling the equation x² = 8 - 5x. Quadratic equations might seem intimidating at first, but with a step-by-step approach, they become much more manageable. In this comprehensive guide, we will explore the methods to solve this equation, understand the underlying concepts, and verify the solutions. So, buckle up and let's get started!

Understanding Quadratic Equations

Before we jump into solving this specific equation, let's briefly discuss what quadratic equations are. 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. The solutions to a quadratic equation are also known as roots or zeros.

Key Characteristics

  • The Highest Power: The highest power of the variable 'x' is 2.
  • Coefficients: The coefficients 'a', 'b', and 'c' determine the shape and position of the parabola when the quadratic equation is graphed.
  • Solutions: A quadratic equation can have two distinct real solutions, one real solution (a repeated root), or two complex solutions.

Why are Quadratic Equations Important?

Quadratic equations appear in various real-world applications, such as physics (projectile motion), engineering (designing structures), and economics (modeling costs and revenues). Mastering the methods to solve them is crucial for anyone studying these fields.

Methods to Solve Quadratic Equations

There are several methods to solve quadratic equations, including:

  1. Factoring: This method involves breaking down the quadratic expression into the product of two binomials.
  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial.
  3. Quadratic Formula: This method uses a formula to directly calculate the solutions.

For the equation x² = 8 - 5x, we will primarily use the quadratic formula, but let's first rearrange the equation into the standard form.

Step-by-Step Solution Using the Quadratic Formula

Step 1: Rearrange the Equation

The first step is to rewrite the equation x² = 8 - 5x in the standard form ax² + bx + c = 0. To do this, we add 5x to both sides and subtract 8 from both sides of the equation:

x² + 5x - 8 = 0

Now, we can identify the coefficients:

  • a = 1
  • b = 5
  • c = -8

Step 2: Apply the Quadratic Formula

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

Now, substitute the values of a, b, and c into the formula:

x = (-5 ± √(5² - 4 * 1 * -8)) / (2 * 1)

Step 3: Simplify the Expression

Let's simplify the expression step by step:

x = (-5 ± √(25 + 32)) / 2

x = (-5 ± √57) / 2

Step 4: Identify the Solutions

So, the two solutions for x are:

  • x₁ = (-5 + √57) / 2
  • x₂ = (-5 - √57) / 2

Thus, the solutions to the equation x² = 8 - 5x are (-5 + √57) / 2 and (-5 - √57) / 2.

Verifying the Solutions

It’s always a good practice to verify the solutions by substituting them back into the original equation. This helps ensure that we didn't make any mistakes during the solving process.

Verifying x₁ = (-5 + √57) / 2

Substitute x₁ into the original equation x² = 8 - 5x:

((-5 + √57) / 2)² = 8 - 5((-5 + √57) / 2)

Expanding and simplifying:

((25 - 10√57 + 57) / 4) = 8 - (-25 + 5√57) / 2

(82 - 10√57) / 4 = (16 + 25 - 5√57) / 2

(41 - 5√57) / 2 = (41 - 5√57) / 2

Since both sides of the equation are equal, x₁ is a valid solution.

Verifying x₂ = (-5 - √57) / 2

Substitute x₂ into the original equation x² = 8 - 5x:

((-5 - √57) / 2)² = 8 - 5((-5 - √57) / 2)

Expanding and simplifying:

((25 + 10√57 + 57) / 4) = 8 - (-25 - 5√57) / 2

(82 + 10√57) / 4 = (16 + 25 + 5√57) / 2

(41 + 5√57) / 2 = (41 + 5√57) / 2

Since both sides of the equation are equal, x₂ is also a valid solution.

Common Mistakes to Avoid

When solving quadratic equations, it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Incorrectly Applying the Quadratic Formula: Ensure you substitute the values of 'a', 'b', and 'c' correctly.
  • Arithmetic Errors: Double-check your calculations, especially when dealing with square roots and fractions.
  • Forgetting to Rearrange the Equation: Always rewrite the equation in the standard form ax² + bx + c = 0 before applying any method.
  • Not Verifying Solutions: Always verify your solutions to catch any mistakes.

Alternative Methods: Factoring and Completing the Square

While we primarily used the quadratic formula, let's briefly touch on the other methods for solving quadratic equations.

Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. This method is efficient when the quadratic expression can be easily factored. However, for equations with irrational or complex roots, factoring can be challenging.

For our equation x² + 5x - 8 = 0, factoring is not straightforward because there are no simple integer factors that multiply to -8 and add up to 5. Hence, we rely on the quadratic formula.

Completing the Square

Completing the square involves transforming the quadratic equation into a perfect square trinomial. This method is useful for deriving the quadratic formula and understanding the structure of quadratic equations.

To complete the square for x² + 5x - 8 = 0:

  1. Move the constant term to the other side: x² + 5x = 8
  2. Add (b/2)² to both sides: x² + 5x + (5/2)² = 8 + (5/2)²
  3. Rewrite as a perfect square: (x + 5/2)² = 8 + 25/4
  4. Simplify: (x + 5/2)² = 57/4
  5. Take the square root of both sides: x + 5/2 = ±√57 / 2
  6. Solve for x: x = -5/2 ± √57 / 2

This gives us the same solutions as the quadratic formula: x = (-5 ± √57) / 2.

Real-World Applications

Understanding and solving quadratic equations is not just an academic exercise; it has practical applications in various fields. Let's look at a couple of examples.

Physics: Projectile Motion

In physics, the trajectory of a projectile (like a ball thrown into the air) can be modeled using a quadratic equation. The equation can help determine the maximum height reached by the projectile and the time it takes to hit the ground.

Engineering: Bridge Design

Engineers use quadratic equations to design structures like bridges. The parabolic shape of suspension cables can be described using a quadratic equation, helping engineers calculate the tension and load distribution.

Economics: Cost and Revenue Analysis

In economics, quadratic equations can be used to model cost and revenue functions. Businesses can use these models to determine the optimal production level to maximize profit.

Conclusion

Solving the quadratic equation x² = 8 - 5x involves rearranging it into standard form and applying the quadratic formula. The solutions we found are x₁ = (-5 + √57) / 2 and x₂ = (-5 - √57) / 2. We verified these solutions by substituting them back into the original equation and also discussed alternative methods like factoring and completing the square. Remember, guys, practice makes perfect, so keep solving those quadratic equations!

Quadratic equations are a fundamental concept in mathematics with far-reaching applications. By understanding the methods to solve them, you're equipping yourself with a powerful tool for problem-solving in various fields. Keep exploring and happy solving!