To provide you with a detailed article and solution, let’s interpret the given keyword “4x 2 5x 12 0” as a polynomial equation that might have a small typo, and it should likely be “4x^2 + 5x – 12 = 0”. Let’s proceed with solving this quadratic equation, offering a step-by-step guide and explanation.

### Understanding Quadratic Equations

A quadratic equation is any equation that can be rearranged in standard form as (ax^2 + bx + c = 0) where (x) represents an unknown variable, and (a), (b), and (c) are constants with (a ≠ 0).

### Problem Statement

Given the quadratic equation (4x^2 + 5x – 12 = 0), find the values of (x).

### Step-by-Step Solution

#### Step 1: Identify the coefficients

For the equation (4x^2 + 5x – 12 = 0), the coefficients are:

- (a = 4)
- (b = 5)
- (c = -12)

#### Step 2: Apply the Quadratic Formula

The solutions for (x) in a quadratic equation can be found using the quadratic formula:

[x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}]

#### Step 3: Substitute the values

Let’s substitute (a), (b), and (c) into the formula:

[x = \frac{-(5) \pm \sqrt{(5)^2 – 4(4)(-12)}}{2(4)}]

#### Step 4: Simplify the expression

We need to calculate the discriminant ((b^2 – 4ac)) and then solve for (x).

Let’s do these calculations.

After calculating, the solutions for the equation (4x^2 + 5x – 12 = 0) are:

[x = -\frac{5}{8} + \frac{\sqrt{217}}{8}]

[x = -\frac{5}{8} – \frac{\sqrt{217}}{8}]

### Conclusion

These solutions mean that the equation (4x^2 + 5x – 12 = 0) has two distinct real roots. The values of (x) are approximately (x ≈ 1.28) and (x ≈ -2.53), which are the points where the quadratic function crosses the x-axis. This equation is an example of how quadratic equations can be solved using the quadratic formula, showcasing the utility of algebra in finding the roots of polynomial equations.