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y^{2}+6y+6=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
y=\frac{-6±\sqrt{6^{2}-4\times 6}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
y=\frac{-6±\sqrt{36-4\times 6}}{2}
Square 6.
y=\frac{-6±\sqrt{36-24}}{2}
Multiply -4 times 6.
y=\frac{-6±\sqrt{12}}{2}
Add 36 to -24.
y=\frac{-6±2\sqrt{3}}{2}
Take the square root of 12.
y=\frac{2\sqrt{3}-6}{2}
Now solve the equation y=\frac{-6±2\sqrt{3}}{2} when ± is plus. Add -6 to 2\sqrt{3}.
y=\sqrt{3}-3
Divide -6+2\sqrt{3} by 2.
y=\frac{-2\sqrt{3}-6}{2}
Now solve the equation y=\frac{-6±2\sqrt{3}}{2} when ± is minus. Subtract 2\sqrt{3} from -6.
y=-\sqrt{3}-3
Divide -6-2\sqrt{3} by 2.
y^{2}+6y+6=\left(y-\left(\sqrt{3}-3\right)\right)\left(y-\left(-\sqrt{3}-3\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3+\sqrt{3} for x_{1} and -3-\sqrt{3} for x_{2}.
x ^ 2 +6x +6 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.
r + s = -6 rs = 6
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -3 - u s = -3 + u
Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-3 - u) (-3 + u) = 6
To solve for unknown quantity u, substitute these in the product equation rs = 6
9 - u^2 = 6
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 6-9 = -3
Simplify the expression by subtracting 9 on both sides
u^2 = 3 u = \pm\sqrt{3} = \pm \sqrt{3}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-3 - \sqrt{3} = -4.732 s = -3 + \sqrt{3} = -1.268
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.