Table of Contents

## Introduction

**Solve: 4x^2-5x-12-0:** Quadratic equations can be challenging to solve, but with the right approach, it’s possible to find the solution. In this guide, we’ll walk you through the steps to solve the equation 4x^2 – 5x – 12 = 0 using methods like factoring and the quadratic formula.

## Understand the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is expressed as x = (-b ± √(b^2 – 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. To use the quadratic formula to solve 4x^2 – 5x – 12 = 0, we first need to identify the values of a, b, and c. In this case, a = 4, b = -5, and c = -12. We can then substitute these values into the quadratic formula and simplify to find the solutions for x.

## Classify the Values of a, b, and c in the Equation

Before we can solve the quadratic equation 4x^2 – 5x – 12 = 0 using the quadratic formula, we need to identify the values of a, b, and c. In this equation, a is the coefficient of the x^2 term, which is 4. B is the coefficient of the x term, which is -5. And c is the constant term, which is -12. Once we have identified these values, we can substitute them into the quadratic formula and solve for x.

## Plug the Values into the Quadratic Formula

Now that we have identified the values of a, b, and c in the quadratic equation 4x^2 – 5x – 12 = 0, we can plug them into the quadratic formula: x = (-b ± √(b^2 – 4ac)) / 2a. Substituting the values, we get x = (-(-5) ± √((-5)^2 – 4(4)(-12))) / 2(4). Simplifying this equation, we get x = (5 ± √241) / 8. These are the two solutions to the quadratic equation 4x^2 – 5x – 12 = 0.

## Simplify the Equation and Solve for x

To solve the quadratic equation 4x^2 – 5x – 12 = 0, we first need to simplify the equation by identifying the values of a, b, and c. In this equation, a = 4, b = -5, and c = -12. Once we have these values, we can plug them into the quadratic formula and solve for x.

**Check your Answer by Plugging into the Original Equation**

After solving a quadratic equation, checking your answer by plugging it back into the original equation is essential. This ensures that the solution is correct and that there are no errors in the calculations. In the case of 4x^2 – 5x – 12 = 0, we would plug the solution for x back into the equation and simplify to see if both are equal. If they are, then we have successfully solved the quadratic equation.

## Alternate Steps to Solve the Quadratic Equation

To solve the quadratic equation 4x^2 – 5x – 12 = 0, we can use the quadratic formula, which is given by:

x = (-b ± sqrt(b^2 – 4ac)) / 2a

Anywhere a, b, and c are the numbers of the quadratic equation. Substituting the values from the given equation, we get:

x = (-(-5) ± sqrt((-5)^2 – 4(4)(-12))) / 2(4)

then,

x = (5 ± sqrt(25 + 192)) / 8

x = (5 ± sqrt(217)) / 8

Therefore, the solutions to the equation 4x^2 – 5x – 12 = 0 are:

x = (5 + sqrt(217)) / 8

and

x = (5 – sqrt(217)) / 8

Which are approximate:

x ≈ 2.116 or x ≈ -1.425