## Solving the Quadratic Equation: (m-2)x² - (5+m)x + 16 = 0

This article explores the solution of the quadratic equation (m-2)x² - (5+m)x + 16 = 0. We'll delve into the different methods for solving it, including the quadratic formula and factoring, and discuss the conditions for the existence of real roots.

### Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

### Solving the Equation

**1. Quadratic Formula:**

The quadratic formula is a powerful tool for solving any quadratic equation. It provides the solutions for x in terms of the coefficients a, b, and c:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, (m-2)x² - (5+m)x + 16 = 0, we have:

- a = m - 2
- b = -(5 + m)
- c = 16

Substituting these values into the quadratic formula, we get:

x = ((5 + m) ± √((-5 - m)² - 4(m - 2)(16))) / 2(m - 2)

This formula provides the two solutions for x.

**2. Factoring:**

Factoring is another way to solve quadratic equations, but it's not always applicable. We need to find two numbers whose product is equal to ac (16(m-2) in our case) and whose sum is equal to b (-5-m). If we can find these numbers, we can rewrite the equation and factor it.

In this case, it might be difficult to find such numbers directly. Therefore, using the quadratic formula is a more reliable approach.

### Conditions for Real Roots

The solutions of a quadratic equation can be real or complex numbers. The nature of the solutions depends on the **discriminant**, which is the expression under the square root in the quadratic formula (b² - 4ac).

**If the discriminant is positive (b² - 4ac > 0), the equation has two distinct real roots.****If the discriminant is zero (b² - 4ac = 0), the equation has one real root (a double root).****If the discriminant is negative (b² - 4ac < 0), the equation has two complex roots.**

For our equation, the discriminant is:

(-5 - m)² - 4(m - 2)(16) = m² + 26m - 109

To determine the conditions for real roots, we need to find the values of m for which the discriminant is greater than or equal to zero:

m² + 26m - 109 ≥ 0

This inequality can be solved by factoring or using the quadratic formula. By solving for m, we can determine the range of values for which the equation has real roots.

### Conclusion

The quadratic equation (m-2)x² - (5+m)x + 16 = 0 can be solved using the quadratic formula. The nature of the solutions depends on the value of the discriminant. Determining the conditions for real roots involves analyzing the discriminant and finding the values of m that satisfy the inequality. This approach provides a complete understanding of the solutions to this quadratic equation.