Solve 5x² - 4x + 3 = 0
5x² - 4x + 3 = 0 has no real solutions. The two complex solutions are x = (2 + √11 i)/5 and x = (2 - √11 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 = -4, 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
\[ 5x^2 - 4x + 3 = 0\ \Rightarrow\ x = \frac{2 + \sqrt{11}\, i}{5},\ \frac{2 - \sqrt{11}\, i}{5} \]
\[ 5x^2 - 4x + 3 = 0 \]
\[ D = (-4)^2 - 4\cdot5\cdot3 = 16 - 60 = -44 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{-44}}{10} \]
\[ x = \frac{2 + \sqrt{11}\, i}{5}\quad\text{or}\quad x = \frac{2 - \sqrt{11}\, i}{5} \]The parabola y = 5x² - 4x + 3
| x | y |
|---|---|
| -2.6 | 47.2 |
| -1.6 | 22.2 |
| -0.6 | 7.2 |
| 0.4 | 2.2 |
| 1.4 | 7.2 |
| 2.4 | 22.2 |
| 3.4 | 47.2 |
Questions
What are the solutions of 5x² - 4x + 3 = 0?
5x² - 4x + 3 = 0 has no real solutions. The two complex solutions are x = (2 + √11 i)/5 and x = (2 - √11 i)/5.
Does 5x² - 4x + 3 = 0 factor?
Not over the integers, because the discriminant -44 is negative. Use the quadratic formula.
What is the vertex of y = 5x² - 4x + 3?
(2/5, 11/5). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 4/5, product = c/a = 3/5.