Solve 3x² - 9x - 3 = 0
The solutions of 3x² - 9x - 3 = 0 are x = (3 + √13)/2 or x = (3 - √13)/2, that is x ≈ 3.3027756 and x ≈ -0.30277564.
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 = 3, 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 ≈ 3.3027756 and x ≈ -0.30277564.
- 7
Check numerically: for x ≈ 3.3027756 the left side is 7.1054274e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 9x - 3 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{13}}{2},\ \frac{3 - \sqrt{13}}{2} \]
\[ 3x^2 - 9x - 3 = 0 \]
\[ D = (-9)^2 - 4\cdot3\cdot(-3) = 81 + 36 = 117 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{117}}{6} \]
\[ x = \frac{3 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{13}}{2} \]The parabola y = 3x² - 9x - 3
| x | y |
|---|---|
| -1.5 | 17.25 |
| -0.5 | 2.25 |
| 0.5 | -6.75 |
| 1.5 | -9.75 |
| 2.5 | -6.75 |
| 3.5 | 2.25 |
| 4.5 | 17.25 |
Questions
What are the solutions of 3x² - 9x - 3 = 0?
The solutions of 3x² - 9x - 3 = 0 are x = (3 + √13)/2 or x = (3 - √13)/2, that is x ≈ 3.3027756 and x ≈ -0.30277564.
Does 3x² - 9x - 3 = 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 = 3x² - 9x - 3?
(3/2, -39/4). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 3, product = c/a = -1.