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

This article will explore the quadratic equation (m-2)x^2 - (5+m)x + 16 = 0, focusing on how to solve it and analyze its solutions based on the value of 'm'.

### Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. It generally takes the form: **ax^2 + bx + c = 0**, where 'a', 'b', and 'c' are coefficients, and 'a' is not equal to zero.

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

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

### Solving the Equation

There are several ways to solve a quadratic equation. Let's examine two common methods:

**1. Quadratic Formula:**
The quadratic formula provides a general solution for any quadratic equation. It states:

**x = (-b ± √(b^2 - 4ac)) / 2a**

Substituting our coefficients, we get:

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

**2. Factoring:**
Factoring is possible when the quadratic equation can be expressed as a product of two linear expressions. This method can be more efficient when applicable.

However, factoring the given equation directly is not straightforward. We need to analyze the discriminant to determine whether factoring is a viable option.

### Analyzing the Discriminant

The discriminant (**Δ**) of a quadratic equation is given by **b^2 - 4ac**. It helps determine the nature of the roots (solutions):

**Δ > 0**: Two distinct real roots. Factoring is possible.**Δ = 0**: One real root (a double root). Factoring is possible as a perfect square.**Δ < 0**: Two complex conjugate roots. Factoring is not possible.

For our equation, **Δ = (-5-m)^2 - 4(m-2)(16)**.

We can simplify this to:

**Δ = m^2 + 18m - 101**

Now, we need to analyze the discriminant based on the value of 'm':

**If m^2 + 18m - 101 > 0**: The equation has two distinct real roots. Factoring is possible if we find the appropriate factors for the quadratic expression.**If m^2 + 18m - 101 = 0**: The equation has one real root. Factoring is possible as a perfect square.**If m^2 + 18m - 101 < 0**: The equation has two complex conjugate roots. Factoring is not possible.

### Conclusion

Solving the quadratic equation (m-2)x^2 - (5+m)x + 16 = 0 involves determining the nature of its roots based on the value of 'm' and applying appropriate methods like the quadratic formula or factoring. The discriminant plays a crucial role in guiding our approach.