Solve -x² - 12x + 11 = 0
The solutions of -x² - 12x + 11 = 0 are x = -6 + √47 or x = -6 - √47, that is x ≈ -12.855655 and x ≈ 0.8556546.
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 = -12, 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 ≈ -12.855655 and x ≈ 0.8556546.
- 7
Check numerically: for x ≈ -12.855655 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 - 12x + 11 = 0\ \Rightarrow\ x = -6 + \sqrt{47},\ -6 - \sqrt{47} \]
\[ -x^2 - 12x + 11 = 0 \]
\[ D = (-12)^2 - 4\cdot(-1)\cdot11 = 144 + 44 = 188 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{188}}{-2} \]
\[ x = -6 + \sqrt{47}\quad\text{or}\quad x = -6 - \sqrt{47} \]The parabola y = -x² - 12x + 11
| x | y |
|---|---|
| -9 | 38 |
| -8 | 43 |
| -7 | 46 |
| -6 | 47 |
| -5 | 46 |
| -4 | 43 |
| -3 | 38 |
Questions
What are the solutions of -x² - 12x + 11 = 0?
The solutions of -x² - 12x + 11 = 0 are x = -6 + √47 or x = -6 - √47, that is x ≈ -12.855655 and x ≈ 0.8556546.
Does -x² - 12x + 11 = 0 factor?
Not over the integers, because the discriminant 188 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -x² - 12x + 11?
(-6, 47). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -12, product = c/a = -11.