Solve -x² - x - 1 = 0
-x² - x - 1 = 0 has no real solutions. The two complex solutions are x = (-1 + √3 i)/2 and x = (-1 - √3 i)/2.
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 = -1, b = -1, c = -1.
- 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
\[ -x^2 - x - 1 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{3}\, i}{2},\ \frac{-1 - \sqrt{3}\, i}{2} \]
\[ -x^2 - x - 1 = 0 \]
\[ D = (-1)^2 - 4\cdot(-1)\cdot(-1) = 1 - 4 = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{-3}}{-2} \]
\[ x = \frac{-1 + \sqrt{3}\, i}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{3}\, i}{2} \]The parabola y = -x² - x - 1
| x | y |
|---|---|
| -3.5 | -9.75 |
| -2.5 | -4.75 |
| -1.5 | -1.75 |
| -0.5 | -0.75 |
| 0.5 | -1.75 |
| 1.5 | -4.75 |
| 2.5 | -9.75 |
Questions
What are the solutions of -x² - x - 1 = 0?
-x² - x - 1 = 0 has no real solutions. The two complex solutions are x = (-1 + √3 i)/2 and x = (-1 - √3 i)/2.
Does -x² - x - 1 = 0 factor?
Not over the integers, because the discriminant -3 is negative. Use the quadratic formula.
What is the vertex of y = -x² - x - 1?
(-1/2, -3/4). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -1, product = c/a = 1.