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x^{2}+14x+22=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.
x=\frac{-14±\sqrt{14^{2}-4\times 22}}{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.
x=\frac{-14±\sqrt{196-4\times 22}}{2}
Square 14.
x=\frac{-14±\sqrt{196-88}}{2}
Multiply -4 times 22.
x=\frac{-14±\sqrt{108}}{2}
Add 196 to -88.
x=\frac{-14±6\sqrt{3}}{2}
Take the square root of 108.
x=\frac{6\sqrt{3}-14}{2}
Now solve the equation x=\frac{-14±6\sqrt{3}}{2} when ± is plus. Add -14 to 6\sqrt{3}.
x=3\sqrt{3}-7
Divide -14+6\sqrt{3} by 2.
x=\frac{-6\sqrt{3}-14}{2}
Now solve the equation x=\frac{-14±6\sqrt{3}}{2} when ± is minus. Subtract 6\sqrt{3} from -14.
x=-3\sqrt{3}-7
Divide -14-6\sqrt{3} by 2.
x^{2}+14x+22=\left(x-\left(3\sqrt{3}-7\right)\right)\left(x-\left(-3\sqrt{3}-7\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7+3\sqrt{3} for x_{1} and -7-3\sqrt{3} for x_{2}.
x ^ 2 +14x +22 = 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 = -14 rs = 22
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 = -7 - u s = -7 + u
Two numbers r and s sum up to -14 exactly when the average of the two numbers is \frac{1}{2}*-14 = -7. 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>
(-7 - u) (-7 + u) = 22
To solve for unknown quantity u, substitute these in the product equation rs = 22
49 - u^2 = 22
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 22-49 = -27
Simplify the expression by subtracting 49 on both sides
u^2 = 27 u = \pm\sqrt{27} = \pm \sqrt{27}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-7 - \sqrt{27} = -12.196 s = -7 + \sqrt{27} = -1.804
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.