Solve 2x² + 9x + 3 = 0
The solutions of 2x² + 9x + 3 = 0 are x = (-9 + √57)/4 or x = (-9 - √57)/4, that is x ≈ -0.36254139 and x ≈ -4.1374586.
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 = 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 ≈ -0.36254139 and x ≈ -4.1374586.
- 7
Check numerically: for x ≈ -0.36254139 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 2x^2 + 9x + 3 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{57}}{4},\ \frac{-9 - \sqrt{57}}{4} \]
\[ 2x^2 + 9x + 3 = 0 \]
\[ D = 9^2 - 4\cdot2\cdot3 = 81 - 24 = 57 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{57}}{4} \]
\[ x = \frac{-9 + \sqrt{57}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{57}}{4} \]The parabola y = 2x² + 9x + 3
| x | y |
|---|---|
| -5.25 | 10.875 |
| -4.25 | 0.875 |
| -3.25 | -5.125 |
| -2.25 | -7.125 |
| -1.25 | -5.125 |
| -0.25 | 0.875 |
| 0.75 | 10.875 |
Questions
What are the solutions of 2x² + 9x + 3 = 0?
The solutions of 2x² + 9x + 3 = 0 are x = (-9 + √57)/4 or x = (-9 - √57)/4, that is x ≈ -0.36254139 and x ≈ -4.1374586.
Does 2x² + 9x + 3 = 0 factor?
Not over the integers, because the discriminant 57 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 2x² + 9x + 3?
(-9/4, -57/8). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -9/2, product = c/a = 3/2.