Solve 5x² + 11x + 3 = 0
The solutions of 5x² + 11x + 3 = 0 are x = (-11 + √61)/10 or x = (-11 - √61)/10, that is x ≈ -0.31897503 and x ≈ -1.881025.
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 = 5, b = 11, 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 ≈ -0.31897503 and x ≈ -1.881025.
- 7
Check numerically: for x ≈ -0.31897503 the left side is -4.4408921e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 5x^2 + 11x + 3 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{61}}{10},\ \frac{-11 - \sqrt{61}}{10} \]
\[ 5x^2 + 11x + 3 = 0 \]
\[ D = 11^2 - 4\cdot5\cdot3 = 121 - 60 = 61 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{61}}{10} \]
\[ x = \frac{-11 + \sqrt{61}}{10}\quad\text{or}\quad x = \frac{-11 - \sqrt{61}}{10} \]The parabola y = 5x² + 11x + 3
| x | y |
|---|---|
| -4.1 | 41.95 |
| -3.1 | 16.95 |
| -2.1 | 1.95 |
| -1.1 | -3.05 |
| -0.1 | 1.95 |
| 0.9 | 16.95 |
| 1.9 | 41.95 |
Questions
What are the solutions of 5x² + 11x + 3 = 0?
The solutions of 5x² + 11x + 3 = 0 are x = (-11 + √61)/10 or x = (-11 - √61)/10, that is x ≈ -0.31897503 and x ≈ -1.881025.
Does 5x² + 11x + 3 = 0 factor?
Not over the integers, because the discriminant 61 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 5x² + 11x + 3?
(-11/10, -61/20). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -11/5, product = c/a = 3/5.