Solve -x² - 4x - 1 = 0
The solutions of -x² - 4x - 1 = 0 are x = -2 + √3 or x = -2 - √3, that is x ≈ -3.7320508 and x ≈ -0.26794919.
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 = -1.
- 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 ≈ -3.7320508 and x ≈ -0.26794919.
- 7
Check numerically: for x ≈ -3.7320508 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 - 4x - 1 = 0\ \Rightarrow\ x = -2 + \sqrt{3},\ -2 - \sqrt{3} \]
\[ -x^2 - 4x - 1 = 0 \]
\[ D = (-4)^2 - 4\cdot(-1)\cdot(-1) = 16 - 4 = 12 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{12}}{-2} \]
\[ x = -2 + \sqrt{3}\quad\text{or}\quad x = -2 - \sqrt{3} \]The parabola y = -x² - 4x - 1
| x | y |
|---|---|
| -5 | -6 |
| -4 | -1 |
| -3 | 2 |
| -2 | 3 |
| -1 | 2 |
| 0 | -1 |
| 1 | -6 |
Questions
What are the solutions of -x² - 4x - 1 = 0?
The solutions of -x² - 4x - 1 = 0 are x = -2 + √3 or x = -2 - √3, that is x ≈ -3.7320508 and x ≈ -0.26794919.
Does -x² - 4x - 1 = 0 factor?
Not over the integers, because the discriminant 12 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -x² - 4x - 1?
(-2, 3). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -4, product = c/a = 1.