Solve 5x² - 10x + 6 = 0
5x² - 10x + 6 = 0 has no real solutions. The two complex solutions are x = (5 + √5 i)/5 and x = (5 - √5 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 = -10, 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
\[ 5x^2 - 10x + 6 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{5}\, i}{5},\ \frac{5 - \sqrt{5}\, i}{5} \]
\[ 5x^2 - 10x + 6 = 0 \]
\[ D = (-10)^2 - 4\cdot5\cdot6 = 100 - 120 = -20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{-20}}{10} \]
\[ x = \frac{5 + \sqrt{5}\, i}{5}\quad\text{or}\quad x = \frac{5 - \sqrt{5}\, i}{5} \]The parabola y = 5x² - 10x + 6
| x | y |
|---|---|
| -2 | 46 |
| -1 | 21 |
| 0 | 6 |
| 1 | 1 |
| 2 | 6 |
| 3 | 21 |
| 4 | 46 |
Questions
What are the solutions of 5x² - 10x + 6 = 0?
5x² - 10x + 6 = 0 has no real solutions. The two complex solutions are x = (5 + √5 i)/5 and x = (5 - √5 i)/5.
Does 5x² - 10x + 6 = 0 factor?
Not over the integers, because the discriminant -20 is negative. Use the quadratic formula.
What is the vertex of y = 5x² - 10x + 6?
(1, 1). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 2, product = c/a = 6/5.