Solve 3x² - 9x + 11 = 0
3x² - 9x + 11 = 0 has no real solutions. The two complex solutions are x = (9 + √51 i)/6 and x = (9 - √51 i)/6.
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 negative, so there are no real solutions; the two solutions are complex numbers.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
LaTeX source of the formulas on this page
\[ 3x^2 - 9x + 11 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{51}\, i}{6},\ \frac{9 - \sqrt{51}\, i}{6} \]
\[ 3x^2 - 9x + 11 = 0 \]
\[ D = (-9)^2 - 4\cdot3\cdot11 = 81 - 132 = -51 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{-51}}{6} \]
\[ x = \frac{9 + \sqrt{51}\, i}{6}\quad\text{or}\quad x = \frac{9 - \sqrt{51}\, i}{6} \]The parabola y = 3x² - 9x + 11
| x | y |
|---|---|
| -1.5 | 31.25 |
| -0.5 | 16.25 |
| 0.5 | 7.25 |
| 1.5 | 4.25 |
| 2.5 | 7.25 |
| 3.5 | 16.25 |
| 4.5 | 31.25 |
Questions
What are the solutions of 3x² - 9x + 11 = 0?
3x² - 9x + 11 = 0 has no real solutions. The two complex solutions are x = (9 + √51 i)/6 and x = (9 - √51 i)/6.
Does 3x² - 9x + 11 = 0 factor?
Not over the integers, because the discriminant -51 is negative. Use the quadratic formula.
What is the vertex of y = 3x² - 9x + 11?
(3/2, 17/4). 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.