Solve 5x² - 12x - 3 = 0
The solutions of 5x² - 12x - 3 = 0 are x = (6 + √51)/5 or x = (6 - √51)/5, that is x ≈ 2.6282857 and x ≈ -0.22828569.
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 = 5, b = -12, c = -3.
- 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 ≈ 2.6282857 and x ≈ -0.22828569.
- 7
Check numerically: for x ≈ 2.6282857 the left side is -3.5527137e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 5x^2 - 12x - 3 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{51}}{5},\ \frac{6 - \sqrt{51}}{5} \]
\[ 5x^2 - 12x - 3 = 0 \]
\[ D = (-12)^2 - 4\cdot5\cdot(-3) = 144 + 60 = 204 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{204}}{10} \]
\[ x = \frac{6 + \sqrt{51}}{5}\quad\text{or}\quad x = \frac{6 - \sqrt{51}}{5} \]The parabola y = 5x² - 12x - 3
| x | y |
|---|---|
| -1.8 | 34.8 |
| -0.8 | 9.8 |
| 0.2 | -5.2 |
| 1.2 | -10.2 |
| 2.2 | -5.2 |
| 3.2 | 9.8 |
| 4.2 | 34.8 |
Questions
What are the solutions of 5x² - 12x - 3 = 0?
The solutions of 5x² - 12x - 3 = 0 are x = (6 + √51)/5 or x = (6 - √51)/5, that is x ≈ 2.6282857 and x ≈ -0.22828569.
Does 5x² - 12x - 3 = 0 factor?
Not over the integers, because the discriminant 204 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 5x² - 12x - 3?
(6/5, -51/5). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 12/5, product = c/a = -3/5.