Solve 5x² - 3x + 11 = 0
5x² - 3x + 11 = 0 has no real solutions. The two complex solutions are x = (3 + √211 i)/10 and x = (3 - √211 i)/10.
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 = -3, c = 11.
- 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
\[ 5x^2 - 3x + 11 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{211}\, i}{10},\ \frac{3 - \sqrt{211}\, i}{10} \]
\[ 5x^2 - 3x + 11 = 0 \]
\[ D = (-3)^2 - 4\cdot5\cdot11 = 9 - 220 = -211 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-211}}{10} \]
\[ x = \frac{3 + \sqrt{211}\, i}{10}\quad\text{or}\quad x = \frac{3 - \sqrt{211}\, i}{10} \]The parabola y = 5x² - 3x + 11
| x | y |
|---|---|
| -2.7 | 55.55 |
| -1.7 | 30.55 |
| -0.7 | 15.55 |
| 0.3 | 10.55 |
| 1.3 | 15.55 |
| 2.3 | 30.55 |
| 3.3 | 55.55 |
Questions
What are the solutions of 5x² - 3x + 11 = 0?
5x² - 3x + 11 = 0 has no real solutions. The two complex solutions are x = (3 + √211 i)/10 and x = (3 - √211 i)/10.
Does 5x² - 3x + 11 = 0 factor?
Not over the integers, because the discriminant -211 is negative. Use the quadratic formula.
What is the vertex of y = 5x² - 3x + 11?
(3/10, 211/20). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 3/5, product = c/a = 11/5.