In mathematics, quadratic equations are a fundamental topic with many applications in science and everyday problem solving. One example is the quadratic equation 4x2 − 5x − 12 = 0. In this article we will solve this equation step by step, explain the methods used, and interpret the results.
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation typically written as ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. For the equation 4x2 − 5x − 12 = 0, the coefficients are:
- a = 4
- b = −5
- c = −12
The goal is to find the values of x that satisfy the equation.
Solving with the quadratic formula
The quadratic formula gives the solutions of ax2 + bx + c = 0 as
x = (−b ± √(b2 − 4ac)) / (2a).
Applying the formula to 4x2 − 5x − 12 = 0 (with a = 4, b = −5, c = −12) gives:
x = (−(−5) ± √((−5)2 − 4·4·(−12))) / (2·4).
Simplifying the expressions inside the square root (the discriminant):
b2 − 4ac = 25 + 192 = 217.
So the exact solutions are
- x₁ = (5 + √217) / 8
- x₂ = (5 − √217) / 8
Because the discriminant (217) is positive, the equation has two distinct real roots. For numerical reference, √217 ≈ 14.7309, which gives the approximations
- x₁ ≈ 2.466
- x₂ ≈ −1.216
Interpreting the solutions
These two values are the x-intercepts of the parabola defined by y = 4x2 − 5x − 12. In other words, they are the x-values where the graph crosses the x-axis. You can visualize the parabola and its roots using an online graphing tool such as Desmos.
If you’d like a deeper explanation of the quadratic formula and the role of the discriminant, see the entries on the quadratic formula and the discriminant. For step-by-step lessons and practice problems, Khan Academy offers helpful tutorials on solving quadratics: Khan Academy — Quadratics.
Conclusion
The quadratic equation 4x2 − 5x − 12 = 0 has two real solutions:
- x₁ = (5 + √217) / 8 ≈ 2.466
- x₂ = (5 − √217) / 8 ≈ −1.216
Solving quadratic equations is a key algebra skill with applications across physics, engineering, and many other fields. The quadratic formula provides a reliable method for finding exact solutions whenever a, b, and c are known.
