Solve x² - x - 1 = 0
The solutions of x² - x - 1 = 0 are x = (1 + √5)/2 or x = (1 - √5)/2, that is x ≈ 1.618034 and x ≈ -0.61803399.
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 positive, so there are two different real solutions.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
- 6
As decimals: x ≈ 1.618034 and x ≈ -0.61803399.
- 7
Check numerically: for x ≈ 1.618034 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 - x - 1 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{5}}{2},\ \frac{1 - \sqrt{5}}{2} \]
\[ x^2 - x - 1 = 0 \]
\[ D = (-1)^2 - 4\cdot1\cdot(-1) = 1 + 4 = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{5}}{2} \]
\[ x = \frac{1 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{5}}{2} \]The parabola y = x² - x - 1
| x | y |
|---|---|
| -2.5 | 7.75 |
| -1.5 | 2.75 |
| -0.5 | -0.25 |
| 0.5 | -1.25 |
| 1.5 | -0.25 |
| 2.5 | 2.75 |
| 3.5 | 7.75 |
Questions
What are the solutions of x² - x - 1 = 0?
The solutions of x² - x - 1 = 0 are x = (1 + √5)/2 or x = (1 - √5)/2, that is x ≈ 1.618034 and x ≈ -0.61803399.
Does x² - x - 1 = 0 factor?
Not over the integers, because the discriminant 5 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² - x - 1?
(1/2, -5/4). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 1, product = c/a = -1.