### 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}]

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