Solve x² - 9x - 9 = 0
The solutions of x² - 9x - 9 = 0 are x = (9 + 3√13)/2 or x = (9 - 3√13)/2, that is x ≈ 9.9083269 and x ≈ -0.90832691.
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 = -9.
- 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 ≈ 9.9083269 and x ≈ -0.90832691.
- 7
Check numerically: for x ≈ 9.9083269 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 - 9x - 9 = 0\ \Rightarrow\ x = \frac{9 + 3\sqrt{13}}{2},\ \frac{9 - 3\sqrt{13}}{2} \]
\[ x^2 - 9x - 9 = 0 \]
\[ D = (-9)^2 - 4\cdot1\cdot(-9) = 81 + 36 = 117 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{117}}{2} \]
\[ x = \frac{9 + 3\sqrt{13}}{2}\quad\text{or}\quad x = \frac{9 - 3\sqrt{13}}{2} \]The parabola y = x² - 9x - 9
| x | y |
|---|---|
| 1.5 | -20.25 |
| 2.5 | -25.25 |
| 3.5 | -28.25 |
| 4.5 | -29.25 |
| 5.5 | -28.25 |
| 6.5 | -25.25 |
| 7.5 | -20.25 |
Questions
What are the solutions of x² - 9x - 9 = 0?
The solutions of x² - 9x - 9 = 0 are x = (9 + 3√13)/2 or x = (9 - 3√13)/2, that is x ≈ 9.9083269 and x ≈ -0.90832691.
Does x² - 9x - 9 = 0 factor?
Not over the integers, because the discriminant 117 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² - 9x - 9?
(9/2, -117/4). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 9, product = c/a = -9.