Solve -2x² + 9x - 12 = 0
-2x² + 9x - 12 = 0 has no real solutions. The two complex solutions are x = (9 + √15 i)/4 and x = (9 - √15 i)/4.
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 = -2, b = 9, c = -12.
- 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
\[ -2x^2 + 9x - 12 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{15}\, i}{4},\ \frac{9 - \sqrt{15}\, i}{4} \]
\[ -2x^2 + 9x - 12 = 0 \]
\[ D = 9^2 - 4\cdot(-2)\cdot(-12) = 81 - 96 = -15 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{-15}}{-4} \]
\[ x = \frac{9 + \sqrt{15}\, i}{4}\quad\text{or}\quad x = \frac{9 - \sqrt{15}\, i}{4} \]The parabola y = -2x² + 9x - 12
| x | y |
|---|---|
| -0.75 | -19.875 |
| 0.25 | -9.875 |
| 1.25 | -3.875 |
| 2.25 | -1.875 |
| 3.25 | -3.875 |
| 4.25 | -9.875 |
| 5.25 | -19.875 |
Questions
What are the solutions of -2x² + 9x - 12 = 0?
-2x² + 9x - 12 = 0 has no real solutions. The two complex solutions are x = (9 + √15 i)/4 and x = (9 - √15 i)/4.
Does -2x² + 9x - 12 = 0 factor?
Not over the integers, because the discriminant -15 is negative. Use the quadratic formula.
What is the vertex of y = -2x² + 9x - 12?
(9/4, -15/8). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 9/2, product = c/a = 6.