Step 1 :The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\). To solve this equation by completing the square, we need to rewrite the equation in the form \((x-h)^2 = k\), where \(h\) and \(k\) are constants. Then, we can solve for \(x\) by taking the square root of both sides of the equation.
Step 2 :The first step is to rewrite the equation in the form \(x^2 + bx = -c\). In this case, we have \(x^2 + 6x = 1\).
Step 3 :Next, we need to complete the square on the left side of the equation. To do this, we add \((b/2)^2\) to both sides of the equation. In this case, \(b = 6\), so we add \((6/2)^2 = 9\) to both sides of the equation. This gives us \((x + 3)^2 = 10\).
Step 4 :Finally, we solve for \(x\) by taking the square root of both sides of the equation. This gives us \(x = -3 \pm \sqrt{10}\).
Step 5 :The solutions to the equation \(x^{2}+6 x-1=0\) are \(x = \boxed{0.16227766016837952}\) and \(x = \boxed{-6.16227766016838}\).