## Introduction

In algebra, quadratic equations play a fundamental role, extending their applications across various disciplines. One such quadratic equation is 4x^2 – 5x – 12 = 0. Solving this equation allows us to determine the values of ‘x’ that satisfy it, known as its roots. Finding these roots involves a combination of methods, such as factoring, completing the square, or using the quadratic formula. This article will delve into these techniques and explore the significance of solving the equation 4x^2 – 5x – 12 = 0.

## Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, typically written in ax^2 + bx + c = 0, where ‘x’ represents the variable, and ‘a’, ‘b,’ and ‘c’ are constants. The highest power of ‘x’ in a quadratic equation is 2, making it a crucial class of polynomial equations.

The equation we will examine in this article is 4x^2 – 5x – 12 = 0. Here, ‘a’ is 4, ‘b’ is -5, and ‘c’ is -12. To solve this equation, we will utilize different techniques, each providing valuable insights into the roots’ nature and the quadratic expression’s behavior.

## Method 1: Factoring the Quadratic Equation

Factoring involves expressing the quadratic equation as a product of two binomials and then setting each binomial to zero to find the roots. For this particular equation, 4x^2 – 5x – 12 = 0, let’s attempt to factor it.

Step 1: Write down the equation 4x^2 – 5x – 12 = 0

Step 2: Find two numbers whose product is ac (4 * -12 = -48) and whose sum is b (-5).

The two numbers that satisfy these conditions are -8 and 3 because (-8) * 3 = -24, and (-8) + 3 = -5.

**Step 3:** Split the middle term of the original equation using the two numbers we found. 4x^2 – 8x + 3x – 12 = 0

Step 4: Factor by grouping. (4x^2 – 8x) + (3x – 12) = 0

Step 5: Factor out the common terms from each group. 4x(x – 2) + 3(x – 4) = 0

Step 6: Set each factor to zero and solve for ‘x.’ 4x – 2 = 0 -> 4x = 2 -> x = 2/4 -> x = 1/2 x – 4 = 0 -> x = 4

The roots of the equation 4x^2 – 5x – 12 = 0 are x = 1/2 and x = 4.

## Method 2: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of ax^2 + bx + c = 0. It is given by:

x = (-b ± √(b^2 – 4ac)) / 2a

Let’s apply the quadratic formula to find the roots of 4x^2 – 5x – 12 = 0.

Step 1: Identify the values of ‘a, ‘b,’ and ‘c.’ In our equation, a = 4, b = -5, and c = -12.

**Step 2:** Plug the values into the quadratic formula. x = (5 ± √((-5)^2 – 4 * 4 * -12)) / 2 * 4

**Step 3:** Simplify the expression under the square root. x = (5 ± √(25 + 192)) / 8 x = (5 ± √217) / 8

**Step 4:** Calculate the roots. x = (5 + √217) / 8 and x = (5 – √217) / 8

The roots of the equation 4x^2 – 5x – 12 = 0 are approximately x = 2.12 and x = -1.12.

## Method 3: Completing the Square

Completing the square is an alternative method to find the roots of a quadratic equation. The process involves transforming the equation into a perfect square trinomial and solving for ‘x.’ Let’s complete the square for our equation, 4x^2 – 5x – 12 = 0.

**Step 1:** Ensure the coefficient of x^2 is 1 (divide the equation by 4). x^2 – (5/4)x – 3 = 0

**Step 2:** Move the constant term to the right side of the equation. x^2 – (5/4)x = 3

**Step 3:** Take half of the coefficient of ‘x’ (5/4) and square it [(5/4)^2 = 25/16]. x^2 – (5/4)x + 25/16 = 3 + 25/16

**Step 4:** Factor the perfect square trinomial on the left side and simplify the constant on the right. (x – 5/4)^2 = 63/16

**Step 5:** Take the square root of both sides. x – 5/4 = ± √(63/16)

Step 6: Solve for ‘x.’ x = 5/4 ± √(63/16)

Step 7: Simplify the expression under the square root. x = 5/4 ± √(63)/4

Step 8: Express the roots separately. x = (5 + √63)/4 and x = (5 – √63)/4

The roots of the equation 4x^2 – 5x – 12 = 0 are approximately x = 2.12 and x = -1.12, which confirms our previous findings.

## Significance of Solving the Quadratic Equation

Quadratic equations have wide-ranging applications in various fields, including physics, engineering, economics, and computer graphics. Some key significance of solving the equation 4x^2 – 5x – 12 = 0 are as follows:

### 1. Finding Intersection Points

In geometry, solving quadratic equations helps find the points where two curves intersect. These points hold importance in understanding the relationship between different elements in the given system.

### 2. Projectile Motion

In physics, quadratic equations play a vital role in describing the trajectory of projectiles. By solving such equations, scientists and engineers can determine a projectile’s maximum height, range, and time of flight.

### 3. Profit and Loss Analysis

In economics and finance, quadratic equations are used to analyze profit and loss scenarios in business. Business owners can find the break-even points and optimize their profits by solving these equations.

### 4. Engineering Applications

Quadratic equations find applications in various engineering problems, such as calculating the stress and strain in structures or analyzing electrical circuits.

### 5. Computer Graphics

In computer graphics, quadratic equations help render curves and surfaces, creating realistic and visually appealing images.

## Conclusion

In conclusion, the quadratic equation 4x^2 – 5x – 12 = 0 holds significant importance in algebra and other fields. By employing methods like factoring, using the quadratic formula, or completing the square, we can find its roots and gain valuable insights into the behavior of the expression. Understanding quadratic equations and their solutions opens the door to solving a wide range of real-world problems and lays the foundation for more advanced mathematical concepts. Whether modeling physical phenomena or optimizing business strategies, the power of quadratic equations is pervasive and enduring.

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