Solve -3x² + 10x - 11 = 0
-3x² + 10x - 11 = 0 has no real solutions. The two complex solutions are x = (5 + 2√2 i)/3 and x = (5 - 2√2 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 = 10, 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
\[ -3x^2 + 10x - 11 = 0\ \Rightarrow\ x = \frac{5 + 2\sqrt{2}\, i}{3},\ \frac{5 - 2\sqrt{2}\, i}{3} \]
\[ -3x^2 + 10x - 11 = 0 \]
\[ D = 10^2 - 4\cdot(-3)\cdot(-11) = 100 - 132 = -32 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{-32}}{-6} \]
\[ x = \frac{5 + 2\sqrt{2}\, i}{3}\quad\text{or}\quad x = \frac{5 - 2\sqrt{2}\, i}{3} \]The parabola y = -3x² + 10x - 11
| x | y |
|---|---|
| -1.3333 | -29.666067 |
| -0.3333 | -14.666267 |
| 0.6667 | -5.6664667 |
| 1.6667 | -2.6666667 |
| 2.6667 | -5.6668667 |
| 3.6667 | -14.667067 |
| 4.6667 | -29.667267 |
Questions
What are the solutions of -3x² + 10x - 11 = 0?
-3x² + 10x - 11 = 0 has no real solutions. The two complex solutions are x = (5 + 2√2 i)/3 and x = (5 - 2√2 i)/3.
Does -3x² + 10x - 11 = 0 factor?
Not over the integers, because the discriminant -32 is negative. Use the quadratic formula.
What is the vertex of y = -3x² + 10x - 11?
(5/3, -8/3). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 10/3, product = c/a = 11/3.