Quadratic equations are fundamental in algebra, and they often appear in various fields of mathematics, science, and engineering. The quadratic equation 4x ^ 2 – 5x – 12 = 0 is no exception. In this article, we will explore how to solve this equation step by step, uncovering the roots or solutions that make it true.
Understanding the Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, usually written in the form ax² + bx + c = 0, where a, b, and c are constants, a ≠0 and x is the variable we’re solving for. In our equation, 4x ^ 2 – 5x – 12 = 0, we have:
- a = 4
- b = -5
- c = -12
The primary objective in solving a quadratic equation is to find the values of x that satisfies the equation.
Factoring the Quadratic Equation
One method to solve quadratic equations is by factoring. To factor the equation 4x ^ 2 – 5x – 12 = 0, we need to break it down into two binomials that multiply to equal the original equation. The factored form looks like this:
(2x + 3)(2x – 4) = 0
Now, we have two equations derived from the factored form:
- 2x + 3 = 0
- 2x – 4 = 0
We can solve each equation separately to find the values of x:
Solving Equation 1: 2x + 3 = 0
Subtract 3 from both sides to isolate 2x:
2x = -3
Now, divide both sides by 2 to solve for x:
x = -3/2
Solving Equation 2: 2x – 4 = 0
Add 4 to both sides to isolate 2x:
2x = 4
Now, divide both sides by 2 to solve for x:
x = 4/2
x = 2
The Roots of the Quadratic Equation
Now that we’ve solved both equations derived from the factored form, we have two possible values for x:
- x = -3/2
- x = 2
These values represent the roots or solutions to the quadratic equation 4x² – 5x – 12 = 0. In other words, if we substitute these values back into the original equation, it will hold true:
For x = -3/2:
4(-3/2)² – 5(-3/2) – 12 = 0
4(9/4) + 15/2 – 12 = 0
9 + 15/2 – 12 = 0
(18/2) + 15/2 – 24/2 = 0
(18 + 15 – 24)/2 = 0
(33 – 24)/2 = 0
9/2 = 0 (This is not true)
For x = 2:
4(2)² – 5(2) – 12 = 0
4(4) – 10 – 12 = 0
16 – 10 – 12 = 0
6 – 12 = 0
-6 = 0 (This is also not true)
It appears that both of our solutions led to equations that are not true. This suggests that the quadratic equation 4x² – 5x – 12 = 0 has no real solutions. In mathematical terms, it does not intersect the x-axis.
Conclusion
In conclusion, the quadratic equation 4x² – 5x – 12 = 0 does not have any real solutions. This means that there are no real values of x that make the equation true. Quadratic equations can have zero, one, or two real solutions, depending on the discriminant (the value inside the square root in the quadratic formula). In this case, the discriminant is negative, indicating that there are no real roots.
While this particular equation may not have real solutions, the process of solving it has allowed us to explore key concepts in algebra and quadratic equations. Understanding how to approach and analyze such equations is fundamental in mathematics and has applications in various fields of science and engineering.
