Welcome to Technical Dhiraj! In today’s lesson, we will walk you through the process of solving the quadratic equation 4x² – 5x – 12 = 0 easily. I
f you’re struggling with quadratic equations, don’t worry. This step-by-step guide will make things clear and simple.
What is a Quadratic Equation?
Before we dive into solving the equation 4x² – 5x – 12 = 0, let’s briefly review what a quadratic equation is.
A quadratic equation is a polynomial equation of degree 2, which means the highest exponent of the variable (usually x) is 2.
The standard form of a quadratic equation is:
ax²+bx+c=0
Where:
- a, b, and c are constants (numbers),
- x is the variable (the unknown you need to solve for),
- And a ≠ 0.
Step 1: Identify the Coefficients in 4x² – 5x – 12 = 0
Let’s begin by identifying the coefficients in the equation 4x² – 5x – 12 = 0:
4x² – 5x – 12 = 0:
- a = 4 (coefficient of x²),
- b = -5 (coefficient of x),
- c = -12 (constant term).
Step 2: Use the Quadratic Formula
The quadratic formula is one of the most common ways to solve quadratic equations. The formula is:
Now, let’s plug in the values of a, b, and c from 4x² – 5x – 12 = 0 into the quadratic formula:
Step 3: Simplify the Expression
Let’s break down the steps to simplify the expression.
Calculate the discriminant (b² – 4ac):
Apply the values in the quadratic formula:
Now, since √217 is not a perfect square, we will leave it as it is or calculate the approximate value.
Step 4: Find the Two Solutions for x
We now have two possible solutions:
First solution:
Second solution:
Thus, the two solutions for the equation 4x² – 5x – 12 = 0 are approximately:
Step 5: Verify the Solutions
To ensure the solutions are correct, you can substitute x = 2.47 and x = -1.22 back into the original equation 4x² – 5x – 12 = 0 and check if both give you zero.
- For x = 2.47, substituting into the equation gives us approximately zero.
- For x = -1.22, substituting into the equation also gives us approximately zero.
This confirms that our solutions are correct!
Conclusion
There you have it! You’ve successfully solved the quadratic equation 4x² – 5x – 12 = 0 using the quadratic formula.
Remember, the quadratic formula is a powerful tool, and by practicing, you’ll become more comfortable with solving similar equations.
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