Solve: If x² – 5x + 6 = 0, find x.

Solving Quadratic Equations: A Step-by-Step Guide

Quadratic equations pop up everywhere, from calculating areas to designing bridges. Let's tackle one together: x² – 5x + 6 = 0

Method 1: Factoring

Factoring breaks down a quadratic equation into simpler expressions. We need to find two numbers that add up to -5 (the coefficient of x) and multiply to 6 (the constant term).

Those numbers are -2 and -3. So, we can rewrite the equation as:

(x - 2)(x - 3) = 0

This means either (x - 2) = 0 or (x - 3) = 0. Therefore, our solutions are:

x = 2 and x = 3

Method 2: Quadratic Formula

The quadratic formula works for all quadratic equations. It's a handy backup if factoring is difficult.

The formula is: x = [-b ± √(b² - 4ac)] / 2a

In our equation, a = 1, b = -5, and c = 6.

Substituting these values:

x = [5 ± √((-5)² - 4 * 1 * 6)] / (2 * 1)

x = [5 ± √(25 - 24)] / 2

x = [5 ± √1] / 2

x = (5 ± 1) / 2

This gives us two solutions:

x = (5 + 1) / 2 = 3

x = (5 - 1) / 2 = 2

Notice that we get the same solutions as with factoring!

Verification

Let's check if our solutions are correct by plugging them back into the original equation:

For x = 2: 2² – 5(2) + 6 = 4 - 10 + 6 = 0 (Correct!)

For x = 3: 3² – 5(3) + 6 = 9 - 15 + 6 = 0 (Correct!)

Conclusion

We've successfully solved the quadratic equation x² – 5x + 6 = 0 using both factoring and the quadratic formula. Both methods yield the solutions x = 2 and x = 3. Knowing multiple methods is key because some equations are easier to solve with one approach than another. Practice makes perfect, so try solving other quadratic equations!