Solve x² - 2x - 1 = 0
The solutions of x² - 2x - 1 = 0 are x = 1 + √2 or x = 1 - √2, that is x ≈ 2.4142136 and x ≈ -0.41421356.
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 = -2, 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 ≈ 2.4142136 and x ≈ -0.41421356.
- 7
Check numerically: for x ≈ 2.4142136 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 - 2x - 1 = 0\ \Rightarrow\ x = 1 + \sqrt{2},\ 1 - \sqrt{2} \]
\[ x^2 - 2x - 1 = 0 \]
\[ D = (-2)^2 - 4\cdot1\cdot(-1) = 4 + 4 = 8 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{8}}{2} \]
\[ x = 1 + \sqrt{2}\quad\text{or}\quad x = 1 - \sqrt{2} \]The parabola y = x² - 2x - 1
| x | y |
|---|---|
| -2 | 7 |
| -1 | 2 |
| 0 | -1 |
| 1 | -2 |
| 2 | -1 |
| 3 | 2 |
| 4 | 7 |
Questions
What are the solutions of x² - 2x - 1 = 0?
The solutions of x² - 2x - 1 = 0 are x = 1 + √2 or x = 1 - √2, that is x ≈ 2.4142136 and x ≈ -0.41421356.
Does x² - 2x - 1 = 0 factor?
Not over the integers, because the discriminant 8 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² - 2x - 1?
(1, -2). 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 = -1.