Solve x² - 11 = 0
The solutions of x² - 11 = 0 are x = √11 or x = -√11, that is x ≈ 3.3166248 and x ≈ -3.3166248.
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 = 0, 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 ≈ 3.3166248 and x ≈ -3.3166248.
- 7
Check numerically: for x ≈ 3.3166248 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 - 11 = 0\ \Rightarrow\ x = \sqrt{11},\ -\sqrt{11} \]
\[ x^2 - 11 = 0 \]
\[ D = 0^2 - 4\cdot1\cdot(-11) = 0 + 44 = 44 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{44}}{2} \]
\[ x = \sqrt{11}\quad\text{or}\quad x = -\sqrt{11} \]The parabola y = x² - 11
| x | y |
|---|---|
| -3 | -2 |
| -2 | -7 |
| -1 | -10 |
| 0 | -11 |
| 1 | -10 |
| 2 | -7 |
| 3 | -2 |
Questions
What are the solutions of x² - 11 = 0?
The solutions of x² - 11 = 0 are x = √11 or x = -√11, that is x ≈ 3.3166248 and x ≈ -3.3166248.
Does x² - 11 = 0 factor?
Not over the integers, because the discriminant 44 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² - 11?
(0, -11). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 0, product = c/a = -11.