Solve 3x² - 6x + 11 = 0
3x² - 6x + 11 = 0 has no real solutions. The two complex solutions are x = (3 + 2√6 i)/3 and x = (3 - 2√6 i)/3.
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 = -6, 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
\[ 3x^2 - 6x + 11 = 0\ \Rightarrow\ x = \frac{3 + 2\sqrt{6}\, i}{3},\ \frac{3 - 2\sqrt{6}\, i}{3} \]
\[ 3x^2 - 6x + 11 = 0 \]
\[ D = (-6)^2 - 4\cdot3\cdot11 = 36 - 132 = -96 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{-96}}{6} \]
\[ x = \frac{3 + 2\sqrt{6}\, i}{3}\quad\text{or}\quad x = \frac{3 - 2\sqrt{6}\, i}{3} \]The parabola y = 3x² - 6x + 11
| x | y |
|---|---|
| -2 | 35 |
| -1 | 20 |
| 0 | 11 |
| 1 | 8 |
| 2 | 11 |
| 3 | 20 |
| 4 | 35 |
Questions
What are the solutions of 3x² - 6x + 11 = 0?
3x² - 6x + 11 = 0 has no real solutions. The two complex solutions are x = (3 + 2√6 i)/3 and x = (3 - 2√6 i)/3.
Does 3x² - 6x + 11 = 0 factor?
Not over the integers, because the discriminant -96 is negative. Use the quadratic formula.
What is the vertex of y = 3x² - 6x + 11?
(1, 8). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 2, product = c/a = 11/3.