Solve -x² - 11x - 3 = 0
The solutions of -x² - 11x - 3 = 0 are x = (-11 + √109)/2 or x = (-11 - √109)/2, that is x ≈ -10.720153 and x ≈ -0.27984675.
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 = -11, c = -3.
- 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 ≈ -10.720153 and x ≈ -0.27984675.
- 7
Check numerically: for x ≈ -10.720153 the left side is -1.4210855e-14 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 - 11x - 3 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{109}}{2},\ \frac{-11 - \sqrt{109}}{2} \]
\[ -x^2 - 11x - 3 = 0 \]
\[ D = (-11)^2 - 4\cdot(-1)\cdot(-3) = 121 - 12 = 109 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{109}}{-2} \]
\[ x = \frac{-11 + \sqrt{109}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{109}}{2} \]The parabola y = -x² - 11x - 3
| x | y |
|---|---|
| -8.5 | 18.25 |
| -7.5 | 23.25 |
| -6.5 | 26.25 |
| -5.5 | 27.25 |
| -4.5 | 26.25 |
| -3.5 | 23.25 |
| -2.5 | 18.25 |
Questions
What are the solutions of -x² - 11x - 3 = 0?
The solutions of -x² - 11x - 3 = 0 are x = (-11 + √109)/2 or x = (-11 - √109)/2, that is x ≈ -10.720153 and x ≈ -0.27984675.
Does -x² - 11x - 3 = 0 factor?
Not over the integers, because the discriminant 109 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -x² - 11x - 3?
(-11/2, 109/4). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -11, product = c/a = 3.