Quadratic equations are fundamental in mathematics, and they appear in various applications across different fields. In this article, we will explore the quadratic equation “4x^2 – 5x – 12 = 0” and demonstrate how to find its solutions.
Understanding the Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, typically represented in the general form: ax^2 + bx + c = 0. In this case, the equation “4x^2 – 5x – 12 = 0” can be written with the coefficients as a = 4, b = -5, and c = -12.
The Quadratic Formula
To find the solutions (roots) of a quadratic equation, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a).
Applying the Quadratic Formula
Now, let’s apply the quadratic formula to the equation “4x^2 – 5x – 12 = 0” to determine its roots:
x = (-(-5) ± √((-5)² – 4(4)(-12))) / (2(4)).
Simplifying further:
x = (5 ± √(25 + 192)) / 8,
x = (5 ± √217) / 8.
We now have two possible solutions:
- x = (5 + √217) / 8
- x = (5 – √217) / 8.
In conclusion, the solutions to the quadratic equation “4x^2 – 5x – 12 = 0” are:
- x = (5 + √217) / 8
- x = (5 – √217) / 8.
These values represent the roots of the equation, and they can be used in various mathematical and practical applications. Quadratic equations are an essential part of algebra, and understanding how to solve them opens up a world of problem-solving possibilities.
Graphical Representation of: 4x^2 – 5x – 12
y= 4x^2 – 5x – 12
Then,
| x | y |
|---|---|
| -3 | 39 |
| -2 | 14 |
| -1 | -3 |
| 0 | -12 |
| 1 | -13 |
| 2 | -6 |
| 3 | 9 |
| 4 | 32 |
| 5 | 63 |
| 6 | 102 |
| 7 | 149 |
