Solve 3x² - 9x - 11 = 0
The solutions of 3x² - 9x - 11 = 0 are x = (9 + √213)/6 or x = (9 - √213)/6, that is x ≈ 3.9324199 and x ≈ -0.93241992.
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 = 3, b = -9, c = -11.
- 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 ≈ 3.9324199 and x ≈ -0.93241992.
- 7
Check numerically: for x ≈ 3.9324199 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 9x - 11 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{213}}{6},\ \frac{9 - \sqrt{213}}{6} \]
\[ 3x^2 - 9x - 11 = 0 \]
\[ D = (-9)^2 - 4\cdot3\cdot(-11) = 81 + 132 = 213 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{213}}{6} \]
\[ x = \frac{9 + \sqrt{213}}{6}\quad\text{or}\quad x = \frac{9 - \sqrt{213}}{6} \]The parabola y = 3x² - 9x - 11
| x | y |
|---|---|
| -1.5 | 9.25 |
| -0.5 | -5.75 |
| 0.5 | -14.75 |
| 1.5 | -17.75 |
| 2.5 | -14.75 |
| 3.5 | -5.75 |
| 4.5 | 9.25 |
Questions
What are the solutions of 3x² - 9x - 11 = 0?
The solutions of 3x² - 9x - 11 = 0 are x = (9 + √213)/6 or x = (9 - √213)/6, that is x ≈ 3.9324199 and x ≈ -0.93241992.
Does 3x² - 9x - 11 = 0 factor?
Not over the integers, because the discriminant 213 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 3x² - 9x - 11?
(3/2, -71/4). 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 = -11/3.