Solve -x² + 10x - 1 = 0
The solutions of -x² + 10x - 1 = 0 are x = 5 + 2√6 or x = 5 - 2√6, that is x ≈ 0.10102051 and x ≈ 9.8989795.
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 = 10, 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 ≈ 0.10102051 and x ≈ 9.8989795.
- 7
Check numerically: for x ≈ 0.10102051 the left side is 4.2188475e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 + 10x - 1 = 0\ \Rightarrow\ x = 5 + 2\sqrt{6},\ 5 - 2\sqrt{6} \]
\[ -x^2 + 10x - 1 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot(-1) = 100 - 4 = 96 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{96}}{-2} \]
\[ x = 5 + 2\sqrt{6}\quad\text{or}\quad x = 5 - 2\sqrt{6} \]The parabola y = -x² + 10x - 1
| x | y |
|---|---|
| 2 | 15 |
| 3 | 20 |
| 4 | 23 |
| 5 | 24 |
| 6 | 23 |
| 7 | 20 |
| 8 | 15 |
Questions
What are the solutions of -x² + 10x - 1 = 0?
The solutions of -x² + 10x - 1 = 0 are x = 5 + 2√6 or x = 5 - 2√6, that is x ≈ 0.10102051 and x ≈ 9.8989795.
Does -x² + 10x - 1 = 0 factor?
Not over the integers, because the discriminant 96 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -x² + 10x - 1?
(5, 24). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 10, product = c/a = 1.