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
\[ -x^2 - 8x - 11 = 0\ \Rightarrow\ x = -4 + \sqrt{5},\ -4 - \sqrt{5} \]
\[ -x^2 - 8x - 11 = 0 \]
\[ D = (-8)^2 - 4\cdot(-1)\cdot(-11) = 64 - 44 = 20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{20}}{-2} \]
\[ x = -4 + \sqrt{5}\quad\text{or}\quad x = -4 - \sqrt{5} \]The parabola y = -x² - 8x - 11
| x | y |
|---|---|
| -7 | -4 |
| -6 | 1 |
| -5 | 4 |
| -4 | 5 |
| -3 | 4 |
| -2 | 1 |
| -1 | -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 downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -8, product = c/a = 11.