Solve x² + 6x + 1 = 0
The solutions of x² + 6x + 1 = 0 are x = -3 + 2√2 or x = -3 - 2√2, that is x ≈ -0.17157288 and x ≈ -5.8284271.
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 = 6, 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 ≈ -0.17157288 and x ≈ -5.8284271.
- 7
Check numerically: for x ≈ -0.17157288 the left side is 1.110223e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 + 6x + 1 = 0\ \Rightarrow\ x = -3 + 2\sqrt{2},\ -3 - 2\sqrt{2} \]
\[ x^2 + 6x + 1 = 0 \]
\[ D = 6^2 - 4\cdot1\cdot1 = 36 - 4 = 32 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{32}}{2} \]
\[ x = -3 + 2\sqrt{2}\quad\text{or}\quad x = -3 - 2\sqrt{2} \]The parabola y = x² + 6x + 1
| x | y |
|---|---|
| -6 | 1 |
| -5 | -4 |
| -4 | -7 |
| -3 | -8 |
| -2 | -7 |
| -1 | -4 |
| 0 | 1 |
Questions
What are the solutions of x² + 6x + 1 = 0?
The solutions of x² + 6x + 1 = 0 are x = -3 + 2√2 or x = -3 - 2√2, that is x ≈ -0.17157288 and x ≈ -5.8284271.
Does x² + 6x + 1 = 0 factor?
Not over the integers, because the discriminant 32 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² + 6x + 1?
(-3, -8). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -6, product = c/a = 1.