Solve -3x² - 6x - 9 = 0
-3x² - 6x - 9 = 0 has no real solutions. The two complex solutions are x = -1 + √2 i and x = -1 - √2 i.
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 = -6, c = -9.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is negative, so there are no real solutions; the two solutions are complex numbers.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
LaTeX source of the formulas on this page
\[ -3x^2 - 6x - 9 = 0\ \Rightarrow\ x = -1 + \sqrt{2}\, i,\ -1 - \sqrt{2}\, i \]
\[ -3x^2 - 6x - 9 = 0 \]
\[ D = (-6)^2 - 4\cdot(-3)\cdot(-9) = 36 - 108 = -72 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{-72}}{-6} \]
\[ x = -1 + \sqrt{2}\, i\quad\text{or}\quad x = -1 - \sqrt{2}\, i \]The parabola y = -3x² - 6x - 9
| x | y |
|---|---|
| -4 | -33 |
| -3 | -18 |
| -2 | -9 |
| -1 | -6 |
| 0 | -9 |
| 1 | -18 |
| 2 | -33 |
Questions
What are the solutions of -3x² - 6x - 9 = 0?
-3x² - 6x - 9 = 0 has no real solutions. The two complex solutions are x = -1 + √2 i and x = -1 - √2 i.
Does -3x² - 6x - 9 = 0 factor?
Not over the integers, because the discriminant -72 is negative. Use the quadratic formula.
What is the vertex of y = -3x² - 6x - 9?
(-1, -6). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -2, product = c/a = 3.