Solve 3x² + 5x + 6 = 0
3x² + 5x + 6 = 0 has no real solutions. The two complex solutions are x = (-5 + √47 i)/6 and x = (-5 - √47 i)/6.
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 = 5, c = 6.
- 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
\[ 3x^2 + 5x + 6 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{47}\, i}{6},\ \frac{-5 - \sqrt{47}\, i}{6} \]
\[ 3x^2 + 5x + 6 = 0 \]
\[ D = 5^2 - 4\cdot3\cdot6 = 25 - 72 = -47 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{-47}}{6} \]
\[ x = \frac{-5 + \sqrt{47}\, i}{6}\quad\text{or}\quad x = \frac{-5 - \sqrt{47}\, i}{6} \]The parabola y = 3x² + 5x + 6
| x | y |
|---|---|
| -3.8333 | 30.916067 |
| -2.8333 | 15.916267 |
| -1.8333 | 6.9164667 |
| -0.8333 | 3.9166667 |
| 0.1667 | 6.9168667 |
| 1.1667 | 15.917067 |
| 2.1667 | 30.917267 |
Questions
What are the solutions of 3x² + 5x + 6 = 0?
3x² + 5x + 6 = 0 has no real solutions. The two complex solutions are x = (-5 + √47 i)/6 and x = (-5 - √47 i)/6.
Does 3x² + 5x + 6 = 0 factor?
Not over the integers, because the discriminant -47 is negative. Use the quadratic formula.
What is the vertex of y = 3x² + 5x + 6?
(-5/6, 47/12). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -5/3, product = c/a = 2.