Solve -3x² + 6x - 5 = 0
-3x² + 6x - 5 = 0 has no real solutions. The two complex solutions are x = (3 + √6 i)/3 and x = (3 - √6 i)/3.
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 = -5.
- 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
\[ -3x^2 + 6x - 5 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{6}\, i}{3},\ \frac{3 - \sqrt{6}\, i}{3} \]
\[ -3x^2 + 6x - 5 = 0 \]
\[ D = 6^2 - 4\cdot(-3)\cdot(-5) = 36 - 60 = -24 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{-24}}{-6} \]
\[ x = \frac{3 + \sqrt{6}\, i}{3}\quad\text{or}\quad x = \frac{3 - \sqrt{6}\, i}{3} \]The parabola y = -3x² + 6x - 5
| x | y |
|---|---|
| -2 | -29 |
| -1 | -14 |
| 0 | -5 |
| 1 | -2 |
| 2 | -5 |
| 3 | -14 |
| 4 | -29 |
Questions
What are the solutions of -3x² + 6x - 5 = 0?
-3x² + 6x - 5 = 0 has no real solutions. The two complex solutions are x = (3 + √6 i)/3 and x = (3 - √6 i)/3.
Does -3x² + 6x - 5 = 0 factor?
Not over the integers, because the discriminant -24 is negative. Use the quadratic formula.
What is the vertex of y = -3x² + 6x - 5?
(1, -2). 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 = 5/3.