Solve: (2x² + 3x – 5 = 0), find x.

Solving Quadratic Equations: A Step-by-Step Guide

Quadratic equations are a fundamental part of algebra, showing up in many areas of math and science. Let's learn how to solve them! We'll tackle the equation 2x² + 3x - 5 = 0 using three common methods.

Method 1: Factoring

Factoring is a great method if it works. We look for two numbers that add up to 3 (the coefficient of x) and multiply to -10 (2 times -5, the product of the coefficient of x² and the constant). Those numbers are 5 and -2. We rewrite the equation as:

2x² + 5x - 2x - 5 = 0

Then we factor by grouping:

x(2x + 5) - 1(2x + 5) = 0

(x - 1)(2x + 5) = 0

This gives us two solutions: x = 1 and x = -5/2

Method 2: Quadratic Formula

The quadratic formula always works, even when factoring is tough or impossible. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

For our equation (2x² + 3x - 5 = 0), a = 2, b = 3, and c = -5. Plugging these values in, we get:

x = [-3 ± √(3² - 4 * 2 * -5)] / (2 * 2)

x = [-3 ± √(49)] / 4

x = [-3 ± 7] / 4

This gives us the same solutions as before: x = 1 and x = -5/2

Method 3: Completing the Square

Completing the square is another powerful technique. It involves manipulating the equation to create a perfect square trinomial. It is more complex than the previous two methods, therefore omitted here for brevity.

Checking Our Solutions

Let's check our solutions (x = 1 and x = -5/2) by plugging them back into the original equation:

For x = 1: 2(1)² + 3(1) - 5 = 0 (Correct!)

For x = -5/2: 2(-5/2)² + 3(-5/2) - 5 = 0 (Correct!)

Conclusion

We've solved the quadratic equation 2x² + 3x - 5 = 0 using factoring and the quadratic formula. Both methods are valuable tools in your algebraic arsenal. Practice makes perfect, so try solving other quadratic equations!