Solve x² - 4x - 11 = 0
The solutions of x² - 4x - 11 = 0 are x = 2 + √15 or x = 2 - √15, that is x ≈ 5.8729833 and x ≈ -1.8729833.
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 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 ≈ 5.8729833 and x ≈ -1.8729833.
- 7
Check numerically: for x ≈ 5.8729833 the left side is 3.5527137e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 - 4x - 11 = 0\ \Rightarrow\ x = 2 + \sqrt{15},\ 2 - \sqrt{15} \]
\[ x^2 - 4x - 11 = 0 \]
\[ D = (-4)^2 - 4\cdot1\cdot(-11) = 16 + 44 = 60 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{60}}{2} \]
\[ x = 2 + \sqrt{15}\quad\text{or}\quad x = 2 - \sqrt{15} \]The parabola y = x² - 4x - 11
| x | y |
|---|---|
| -1 | -6 |
| 0 | -11 |
| 1 | -14 |
| 2 | -15 |
| 3 | -14 |
| 4 | -11 |
| 5 | -6 |
Questions
What are the solutions of x² - 4x - 11 = 0?
The solutions of x² - 4x - 11 = 0 are x = 2 + √15 or x = 2 - √15, that is x ≈ 5.8729833 and x ≈ -1.8729833.
Does x² - 4x - 11 = 0 factor?
Not over the integers, because the discriminant 60 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² - 4x - 11?
(2, -15). 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.