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 ≈ -0.61803399 and x ≈ 1.618034.
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 ≈ -0.61803399 and x ≈ 1.618034.
- 7
Check numerically: for x ≈ -0.61803399 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 + x + 1 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{5}}{2},\ \frac{1 - \sqrt{5}}{2} \]
\[ -x^2 + x + 1 = 0 \]
\[ D = 1^2 - 4\cdot(-1)\cdot1 = 1 + 4 = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{5}}{-2} \]
\[ x = \frac{1 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{5}}{2} \]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 ≈ -0.61803399 and x ≈ 1.618034.
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 downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 1, product = c/a = -1.