Solve -3x² - 2x - 11 = 0
-3x² - 2x - 11 = 0 has no real solutions. The two complex solutions are x = (-1 + 4√2 i)/3 and x = (-1 - 4√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 = -2, 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
\[ -3x^2 - 2x - 11 = 0\ \Rightarrow\ x = \frac{-1 + 4\sqrt{2}\, i}{3},\ \frac{-1 - 4\sqrt{2}\, i}{3} \]
\[ -3x^2 - 2x - 11 = 0 \]
\[ D = (-2)^2 - 4\cdot(-3)\cdot(-11) = 4 - 132 = -128 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-128}}{-6} \]
\[ x = \frac{-1 + 4\sqrt{2}\, i}{3}\quad\text{or}\quad x = \frac{-1 - 4\sqrt{2}\, i}{3} \]The parabola y = -3x² - 2x - 11
| x | y |
|---|---|
| -3.3333 | -37.666067 |
| -2.3333 | -22.666267 |
| -1.3333 | -13.666467 |
| -0.3333 | -10.666667 |
| 0.6667 | -13.666867 |
| 1.6667 | -22.667067 |
| 2.6667 | -37.667267 |
Questions
What are the solutions of -3x² - 2x - 11 = 0?
-3x² - 2x - 11 = 0 has no real solutions. The two complex solutions are x = (-1 + 4√2 i)/3 and x = (-1 - 4√2 i)/3.
Does -3x² - 2x - 11 = 0 factor?
Not over the integers, because the discriminant -128 is negative. Use the quadratic formula.
What is the vertex of y = -3x² - 2x - 11?
(-1/3, -32/3). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -2/3, product = c/a = 11/3.