Solving Quadratic Equations: A Step-by-Step Guide to 4x^2 – 5x – 12 = 0

Quadratic equations are an essential part of algebra and often appear in various mathematical and scientific applications. These equations are ax^2 + bx + c = 0, where a, b, and c are constants, and x is the unknown variable. This article will explore how to solve the quadratic equation 4x^2 – 5x – 12 = 0 using various methods, providing a comprehensive step-by-step guide to tackle such equations confidently.

Understanding Quadratic Equations

Before diving into the solution, it is crucial to understand what a quadratic equation represents. A quadratic equation is a polynomial equation of degree 2, meaning the variable’s highest power (x in this case) is 2. The general form is ax^2 + bx + c = 0, where ‘a,’ ‘b,’ and ‘c’ are real numbers, and ‘a’ should not equal zero. Solving the quadratic equation involves finding the value(s) of ‘x’ that satisfies the equation.

Factoring the Quadratic Equation

One of the most common methods to solve a quadratic equation is factoring. In this method, we express the equation as a product of two binomials. For example, for the given equation 4x^2 – 5x – 12 = 0, we need to find two binomials whose product equals the equation. The equation can be factored as (4x + 3)(x – 4) = 0.

To find the values of ‘x,’ we set each factor to zero and solve for ‘x’:

1. 4x + 3 = 0: Subtract three from both sides and divide by 4.

• • 4x = -3
  • x = -3/4

1. x – 4 = 0: Add 4 to both sides.

• • x = 4

Therefore, the solutions for the quadratic equation 4x^2 – 5x – 12 = 0 are x = -3/4 and x = 4.

Using the Quadratic Formula

Another method to solve quadratic equations is by using the quadratic formula. The quadratic formula states that for any quadratic equation ax^2 + bx + c = 0, the solutions for ‘x’ are given by:

x = (-b ± √(b^2 – 4ac)) / 2a

Let’s apply the quadratic formula to our equation 4x^2 – 5x – 12 = 0:

1. Identify the values of ‘a,’ ‘b,’ and ‘c’:

• • a = 4
  • b = -5
  • c = -12

1. Plug the values into the quadratic formula:

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

1. Calculate the discriminant (the value inside the square root):

Discriminant = (-5)^2 – 4 * 4 * -12 = 25 + 192 = 217

1. Continue solving for ‘x’:

x = (5 ± √217) / 8

Since the discriminant is positive (√217 is a real number), the equation has two real solutions:

x = (5 + √217) / 8 ≈ 2.219 x = (5 – √217) / 8 ≈ -0.719

Graphical Representation of Solutions

We can also visualize the solutions to the quadratic equation using a graph. Plotting the graph of the equation y = 4x^2 – 5x – 12, we can see the points where the graph intersects the x-axis are the solutions to the equation.

The graph shows two points of intersection at x ≈ 2.219 and x ≈ -0.719, confirming the solutions we obtained through factoring and using the quadratic formula.

Completing the Square Method

An alternative method to solve quadratic equations is completing the square procedure. This method involves rewriting the equation in a perfect fair form and solving for ‘x.’

1. Start with the quadratic equation 4x^2 – 5x – 12 = 0.
2. Move the constant term (-12) to the other side of the equation:

4x^2 – 5x = 12

1. To complete the square, add and subtract the square of half of the coefficient of ‘x’ ((-5)/2)^2 = 25/4:

4x^2 – 5x + 25/4 – 25/4 = 12

1. Factor the perfect square trinomial:

(2x – 5/2)^2 = 49/4

1. Take the square root of both sides and solve for ‘x’:

2x – 5/2 = ±√(49/4)

2x – 5/2 = ±7/2

1. Solve for ‘x’:https://techhubinfo.com/2022/09/19/factors-to-consider-if-you-really-want-to-decorate-your-kitchen-perfectly/

a. 2x – 5/2 = 7/2 2x = 7/2 + 5/2 2x = 12/2 x = 6/2 x = 3

b. 2x – 5/2 = -7/2 2x = -7/2 + 5/2 2x = -2/2 x = -1

Thus, using the completing square method, the solutions to the quadratic equation 4x^2 – 5x – 12 = 0 are x = 3 and x = -1.

Conclusion

Solving quadratic equations is an essential skill in mathematics and various real-world applications. This article explored several methods to solve the quadratic equation 4x^2 – 5x – 12 = 0, including factoring, using the quadratic formula, and completing the square procedure. Each method provides a unique approach to finding the solutions, and understanding these techniques will enhance your problem-solving abilities in mathematics.

FAQs

1. What is a quadratic equation? A quadratic equation is a polynomial equation of degree 2, represented as ax^2 + bx + c = 0, where ‘a,’ ‘b,’ and ‘c’ are constants, and ‘x’ is the variable.
2. What are the methods to solve quadratic equations? Several ways exist to solve quadratic equations, including factoring, using the quadratic formula, and completing the square procedure.
3. Can all quadratic equations be factored in? No, not all quadratic equations can be factored. Some equations may require the quadratic formula or completing the square method to find the solutions.