Solve The Equation For X 5x 2 4x 6

Solving the Equation 5x² + 4x = 6: A Comprehensive Guide

Solving algebraic equations is a fundamental skill in mathematics, crucial for various applications in science, engineering, and everyday life. This article delves into the solution of the quadratic equation 5x² + 4x = 6, exploring different methods and providing a detailed explanation of each step. We'll cover the quadratic formula, factoring, and completing the square, emphasizing the importance of understanding the underlying principles and choosing the most efficient method. We'll also discuss checking solutions and interpreting the results.

Understanding Quadratic Equations

Before we dive into solving the equation, let's review the basics of quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our equation, 5x² + 4x = 6, needs to be rearranged into this standard form before we can apply solution methods.

Rearranging the Equation

The first step is to rewrite the equation in the standard form ax² + bx + c = 0. To do this, we subtract 6 from both sides of the equation:

5x² + 4x - 6 = 0

Now we have our equation in the standard quadratic form, with a = 5, b = 4, and c = -6. We can now proceed to solve for 'x' using various methods.

Method 1: Using the Quadratic Formula

The quadratic formula is a powerful tool that provides a direct solution for any quadratic equation. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 5, b = 4, c = -6), we get:

x = [-4 ± √(4² - 4 * 5 * -6)] / (2 * 5)

Let's break down the calculation step-by-step:

1. Calculate the discriminant (b² - 4ac): 4² - 4 * 5 * -6 = 16 + 120 = 136

2. Substitute into the quadratic formula: x = [-4 ± √136] / 10

3. Simplify the square root: √136 = √(4 * 34) = 2√34

4. Final solutions: x = (-4 + 2√34) / 10 and x = (-4 - 2√34) / 10

These can be further simplified to:

x = (-2 + √34) / 5 and x = (-2 - √34) / 5

These are the exact solutions. Using a calculator, we can find approximate decimal values:

x ≈ 0.824 and x ≈ -1.444

Method 2: Factoring

Factoring is a method that involves expressing the quadratic equation as a product of two linear factors. This method is only applicable if the quadratic expression can be easily factored. In this case, factoring our equation (5x² + 4x - 6 = 0) directly is not straightforward because there aren't readily apparent integer factors that multiply to -30 (5 * -6) and add to 4.

Method 3: Completing the Square

Completing the square is another powerful technique to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

1. Divide by 'a': Divide the entire equation by 5: x² + (4/5)x - (6/5) = 0

2. Move the constant to the right side: x² + (4/5)x = 6/5

3. Find the value to complete the square: Take half of the coefficient of x (4/5), square it, and add it to both sides: (4/5) / 2 = 2/5 (2/5)² = 4/25 x² + (4/5)x + 4/25 = 6/5 + 4/25

4. Factor the perfect square trinomial: (x + 2/5)² = 34/25

5. Solve for x: x + 2/5 = ±√(34/25) x = -2/5 ± (√34)/5

This gives us the same solutions as the quadratic formula:

x = (-2 + √34) / 5 and x = (-2 - √34) / 5

Checking the Solutions

It's crucial to check our solutions by substituting them back into the original equation:

For x ≈ 0.824:

5(0.824)² + 4(0.824) ≈ 3.4 + 3.296 ≈ 6.696 (approximately 6 due to rounding)

For x ≈ -1.444:

5(-1.444)² + 4(-1.444) ≈ 10.46 - 5.776 ≈ 4.684 (approximately 6 due to rounding)

The slight discrepancies are due to rounding errors in the approximate decimal values. Substituting the exact solutions will yield precisely 6 in both cases.

Choosing the Right Method

The best method for solving a quadratic equation depends on the specific equation. The quadratic formula is always applicable and provides exact solutions. Factoring is the quickest method if the equation is easily factorable. Completing the square is a useful technique for understanding the underlying structure of quadratic equations and can be helpful in other areas of mathematics.

Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have widespread real-world applications. Here are a few examples:

• Physics: Calculating projectile motion, determining the trajectory of an object under the influence of gravity.
• Engineering: Designing bridges, buildings, and other structures.
• Economics: Modeling supply and demand, determining optimal production levels.
• Computer graphics: Creating curves and shapes in computer-aided design.

Conclusion

Solving the equation 5x² + 4x = 6 demonstrates the versatility of different algebraic techniques. While the quadratic formula provides a direct solution, understanding factoring and completing the square provides deeper insights into the structure and properties of quadratic equations. Remember to always check your solutions and choose the most efficient method depending on the specific problem at hand. The ability to solve quadratic equations is a fundamental skill with far-reaching implications across various fields. Mastering these techniques opens doors to solving more complex mathematical problems and applying them to real-world situations.