Solve 5x² - x - 1 = 0
The solutions of 5x² - x - 1 = 0 are x = (1 + √21)/10 or x = (1 - √21)/10, that is x ≈ 0.55825757 and x ≈ -0.35825757.
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 = 5, 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 ≈ 0.55825757 and x ≈ -0.35825757.
- 7
Check numerically: for x ≈ 0.55825757 the left side is 2.220446e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 5x^2 - x - 1 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{21}}{10},\ \frac{1 - \sqrt{21}}{10} \]
\[ 5x^2 - x - 1 = 0 \]
\[ D = (-1)^2 - 4\cdot5\cdot(-1) = 1 + 20 = 21 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{21}}{10} \]
\[ x = \frac{1 + \sqrt{21}}{10}\quad\text{or}\quad x = \frac{1 - \sqrt{21}}{10} \]The parabola y = 5x² - x - 1
| x | y |
|---|---|
| -2.9 | 43.95 |
| -1.9 | 18.95 |
| -0.9 | 3.95 |
| 0.1 | -1.05 |
| 1.1 | 3.95 |
| 2.1 | 18.95 |
| 3.1 | 43.95 |
Questions
What are the solutions of 5x² - x - 1 = 0?
The solutions of 5x² - x - 1 = 0 are x = (1 + √21)/10 or x = (1 - √21)/10, that is x ≈ 0.55825757 and x ≈ -0.35825757.
Does 5x² - x - 1 = 0 factor?
Not over the integers, because the discriminant 21 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 5x² - x - 1?
(1/10, -21/20). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 1/5, product = c/a = -1/5.