Solve -x² + 9x - 3 = 0
The solutions of -x² + 9x - 3 = 0 are x = (9 + √69)/2 or x = (9 - √69)/2, that is x ≈ 0.34668807 and x ≈ 8.6533119.
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 = 9, c = -3.
- 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 ≈ 0.34668807 and x ≈ 8.6533119.
- 7
Check numerically: for x ≈ 0.34668807 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 + 9x - 3 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{69}}{2},\ \frac{9 - \sqrt{69}}{2} \]
\[ -x^2 + 9x - 3 = 0 \]
\[ D = 9^2 - 4\cdot(-1)\cdot(-3) = 81 - 12 = 69 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{69}}{-2} \]
\[ x = \frac{9 + \sqrt{69}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{69}}{2} \]The parabola y = -x² + 9x - 3
| x | y |
|---|---|
| 1.5 | 8.25 |
| 2.5 | 13.25 |
| 3.5 | 16.25 |
| 4.5 | 17.25 |
| 5.5 | 16.25 |
| 6.5 | 13.25 |
| 7.5 | 8.25 |
Questions
What are the solutions of -x² + 9x - 3 = 0?
The solutions of -x² + 9x - 3 = 0 are x = (9 + √69)/2 or x = (9 - √69)/2, that is x ≈ 0.34668807 and x ≈ 8.6533119.
Does -x² + 9x - 3 = 0 factor?
Not over the integers, because the discriminant 69 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -x² + 9x - 3?
(9/2, 69/4). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 9, product = c/a = 3.