Solve 4x² - 12x - 5 = 0
The solutions of 4x² - 12x - 5 = 0 are x = (3 + √14)/2 or x = (3 - √14)/2, that is x ≈ 3.3708287 and x ≈ -0.37082869.
Change a, b or c (whole numbers, a not 0) and the exact solutions follow.
How to work it out
- 1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = -12, c = -5.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is positive, so there are two different real solutions.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
- 6
As decimals: x ≈ 3.3708287 and x ≈ -0.37082869.
- 7
Check numerically: for x ≈ 3.3708287 the left side is -7.1054274e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 4x^2 - 12x - 5 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{14}}{2},\ \frac{3 - \sqrt{14}}{2} \]
\[ 4x^2 - 12x - 5 = 0 \]
\[ D = (-12)^2 - 4\cdot4\cdot(-5) = 144 + 80 = 224 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{224}}{8} \]
\[ x = \frac{3 + \sqrt{14}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{14}}{2} \]The parabola y = 4x² - 12x - 5
| x | y |
|---|---|
| -1.5 | 22 |
| -0.5 | 2 |
| 0.5 | -10 |
| 1.5 | -14 |
| 2.5 | -10 |
| 3.5 | 2 |
| 4.5 | 22 |
Questions
What are the solutions of 4x² - 12x - 5 = 0?
The solutions of 4x² - 12x - 5 = 0 are x = (3 + √14)/2 or x = (3 - √14)/2, that is x ≈ 3.3708287 and x ≈ -0.37082869.
Does 4x² - 12x - 5 = 0 factor?
Not over the integers, because the discriminant 224 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 4x² - 12x - 5?
(3/2, -14). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 3, product = c/a = -5/4.