Solve The Equation For X 5x 2 4x 6

Solving the Equation: 5x² + 4x = 6

This comprehensive guide will walk you through various methods of solving the quadratic equation 5x² + 4x = 6. We'll explore different approaches, highlighting their strengths and weaknesses, and providing a step-by-step explanation for each. Understanding quadratic equations is crucial in algebra and numerous applications in science, engineering, and finance. Let's dive in!

Understanding Quadratic Equations

Before we begin solving, let's establish a firm understanding of what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this 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 zero (otherwise, it wouldn't be a quadratic equation). Our equation, 5x² + 4x = 6, needs to be rearranged into this standard form before we can effectively solve it.

Rearranging the Equation

The first step in solving our equation is to rearrange it into 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.

Method 1: Factoring

Factoring is a method used to find the roots (solutions) of a quadratic equation by expressing it as a product of two linear factors. Unfortunately, this method isn't always straightforward and sometimes doesn't work for all quadratic equations. Let's see if it works for our equation:

We're looking for two numbers that add up to 4 (the coefficient of x) and multiply to -30 (the product of a and c, which is 5 * -6). Unfortunately, there are no two integers that satisfy these conditions. This means our equation is not easily factorable using integer coefficients. Therefore, we'll need to explore other methods.

Method 2: Quadratic Formula

The quadratic formula is a powerful tool that provides a direct solution for any quadratic equation, regardless of whether it's easily factorable or not. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Let's plug in the values from our equation (a = 5, b = 4, c = -6):

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

x = [-4 ± √(16 + 120)] / 10

x = [-4 ± √136] / 10

Now, we can simplify √136: √136 = √(4 * 34) = 2√34

Therefore, our solutions are:

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 to our quadratic equation. We can approximate these values using a calculator:

x ≈ 0.87 and x ≈ -1.37

Method 3: Completing the Square

Completing the square is another method for solving quadratic equations. This method 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 ((4/5)/2)² = (2/5)² = 4/25, and add it to both sides: 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

Method 4: Graphical Method

A graphical method can also be used to solve the quadratic equation. By plotting the graph of the function y = 5x² + 4x - 6, the x-intercepts represent the solutions to the equation (where y = 0). You can use graphing software or a graphing calculator to plot the function and find the x-intercepts, which will approximate the solutions we obtained earlier.

Comparing Methods

Each method has its advantages and disadvantages:

• Factoring: Simplest when applicable but not always possible.
• Quadratic Formula: Always works but can be more computationally intensive.
• Completing the Square: A systematic approach, useful for understanding the structure of quadratic equations and deriving the quadratic formula.
• Graphical Method: Provides a visual representation of the solutions, but relies on accurate plotting and may not give exact solutions.

Applications of Quadratic Equations

Quadratic equations appear in many real-world scenarios:

• Physics: Projectile motion, calculating the trajectory of objects under gravity.
• Engineering: Designing structures, optimizing shapes for strength and efficiency.
• Finance: Modeling compound interest, determining investment returns.
• Economics: Analyzing supply and demand curves, maximizing profits.

Conclusion

Solving the quadratic equation 5x² + 4x = 6 demonstrates the versatility of different algebraic techniques. While factoring isn't always feasible, the quadratic formula provides a reliable solution. Completing the square offers a valuable understanding of quadratic equation structure. The graphical method provides visual insight. Choosing the most appropriate method depends on the specific equation and the desired level of precision. Understanding these methods equips you with powerful tools for tackling numerous mathematical problems in various fields. Remember to always check your solutions by substituting them back into the original equation to ensure accuracy.