Solve x² - 4x + 11 = 0
x² - 4x + 11 = 0 has no real solutions. The two complex solutions are x = 2 + √7 i and x = 2 - √7 i.
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 = 1, b = -4, 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
\[ x^2 - 4x + 11 = 0\ \Rightarrow\ x = 2 + \sqrt{7}\, i,\ 2 - \sqrt{7}\, i \]
\[ x^2 - 4x + 11 = 0 \]
\[ D = (-4)^2 - 4\cdot1\cdot11 = 16 - 44 = -28 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{-28}}{2} \]
\[ x = 2 + \sqrt{7}\, i\quad\text{or}\quad x = 2 - \sqrt{7}\, i \]The parabola y = x² - 4x + 11
| x | y |
|---|---|
| -1 | 16 |
| 0 | 11 |
| 1 | 8 |
| 2 | 7 |
| 3 | 8 |
| 4 | 11 |
| 5 | 16 |
Questions
What are the solutions of x² - 4x + 11 = 0?
x² - 4x + 11 = 0 has no real solutions. The two complex solutions are x = 2 + √7 i and x = 2 - √7 i.
Does x² - 4x + 11 = 0 factor?
Not over the integers, because the discriminant -28 is negative. Use the quadratic formula.
What is the vertex of y = x² - 4x + 11?
(2, 7). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 4, product = c/a = 11.