Solve 5x² + 2x + 3 = 0
5x² + 2x + 3 = 0 has no real solutions. The two complex solutions are x = (-1 + √14 i)/5 and x = (-1 - √14 i)/5.
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 = 2, c = 3.
- 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
\[ 5x^2 + 2x + 3 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{14}\, i}{5},\ \frac{-1 - \sqrt{14}\, i}{5} \]
\[ 5x^2 + 2x + 3 = 0 \]
\[ D = 2^2 - 4\cdot5\cdot3 = 4 - 60 = -56 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{-56}}{10} \]
\[ x = \frac{-1 + \sqrt{14}\, i}{5}\quad\text{or}\quad x = \frac{-1 - \sqrt{14}\, i}{5} \]The parabola y = 5x² + 2x + 3
| x | y |
|---|---|
| -3.2 | 47.8 |
| -2.2 | 22.8 |
| -1.2 | 7.8 |
| -0.2 | 2.8 |
| 0.8 | 7.8 |
| 1.8 | 22.8 |
| 2.8 | 47.8 |
Questions
What are the solutions of 5x² + 2x + 3 = 0?
5x² + 2x + 3 = 0 has no real solutions. The two complex solutions are x = (-1 + √14 i)/5 and x = (-1 - √14 i)/5.
Does 5x² + 2x + 3 = 0 factor?
Not over the integers, because the discriminant -56 is negative. Use the quadratic formula.
What is the vertex of y = 5x² + 2x + 3?
(-1/5, 14/5). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -2/5, product = c/a = 3/5.