Solve 3x² - x - 1 = 0
The solutions of 3x² - x - 1 = 0 are x = (1 + √13)/6 or x = (1 - √13)/6, that is x ≈ 0.76759188 and x ≈ -0.43425855.
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 = -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 ≈ 0.76759188 and x ≈ -0.43425855.
- 7
Check numerically: for x ≈ 0.76759188 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 - x - 1 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{13}}{6},\ \frac{1 - \sqrt{13}}{6} \]
\[ 3x^2 - x - 1 = 0 \]
\[ D = (-1)^2 - 4\cdot3\cdot(-1) = 1 + 12 = 13 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{13}}{6} \]
\[ x = \frac{1 + \sqrt{13}}{6}\quad\text{or}\quad x = \frac{1 - \sqrt{13}}{6} \]The parabola y = 3x² - x - 1
| x | y |
|---|---|
| -2.8333 | 25.916067 |
| -1.8333 | 10.916267 |
| -0.8333 | 1.9164667 |
| 0.1667 | -1.0833333 |
| 1.1667 | 1.9168667 |
| 2.1667 | 10.917067 |
| 3.1667 | 25.917267 |
Questions
What are the solutions of 3x² - x - 1 = 0?
The solutions of 3x² - x - 1 = 0 are x = (1 + √13)/6 or x = (1 - √13)/6, that is x ≈ 0.76759188 and x ≈ -0.43425855.
Does 3x² - x - 1 = 0 factor?
Not over the integers, because the discriminant 13 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 3x² - x - 1?
(1/6, -13/12). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 1/3, product = c/a = -1/3.