Solve x² - 8x + 11 = 0
The solutions of x² - 8x + 11 = 0 are x = 4 + √5 or x = 4 - √5, that is x ≈ 6.236068 and x ≈ 1.763932.
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 = -8, c = 11.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is positive, so there are two different real solutions.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
- 6
As decimals: x ≈ 6.236068 and x ≈ 1.763932.
- 7
Check numerically: for x ≈ 6.236068 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 - 8x + 11 = 0\ \Rightarrow\ x = 4 + \sqrt{5},\ 4 - \sqrt{5} \]
\[ x^2 - 8x + 11 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot11 = 64 - 44 = 20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{20}}{2} \]
\[ x = 4 + \sqrt{5}\quad\text{or}\quad x = 4 - \sqrt{5} \]The parabola y = x² - 8x + 11
| x | y |
|---|---|
| 1 | 4 |
| 2 | -1 |
| 3 | -4 |
| 4 | -5 |
| 5 | -4 |
| 6 | -1 |
| 7 | 4 |
Questions
What are the solutions of x² - 8x + 11 = 0?
The solutions of x² - 8x + 11 = 0 are x = 4 + √5 or x = 4 - √5, that is x ≈ 6.236068 and x ≈ 1.763932.
Does x² - 8x + 11 = 0 factor?
Not over the integers, because the discriminant 20 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² - 8x + 11?
(4, -5). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 8, product = c/a = 11.