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2x^{2}+4x+1=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{-4±\sqrt{4^{2}-4\times 2}}{2\times 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{-4±\sqrt{16-4\times 2}}{2\times 2}
Square 4.
x=\frac{-4±\sqrt{16-8}}{2\times 2}
Multiply -4 times 2.
x=\frac{-4±\sqrt{8}}{2\times 2}
Add 16 to -8.
x=\frac{-4±2\sqrt{2}}{2\times 2}
Take the square root of 8.
x=\frac{-4±2\sqrt{2}}{4}
Multiply 2 times 2.
x=\frac{2\sqrt{2}-4}{4}
Now solve the equation x=\frac{-4±2\sqrt{2}}{4} when ± is plus. Add -4 to 2\sqrt{2}.
x=\frac{\sqrt{2}}{2}-1
Divide -4+2\sqrt{2} by 4.
x=\frac{-2\sqrt{2}-4}{4}
Now solve the equation x=\frac{-4±2\sqrt{2}}{4} when ± is minus. Subtract 2\sqrt{2} from -4.
x=-\frac{\sqrt{2}}{2}-1
Divide -4-2\sqrt{2} by 4.
2x^{2}+4x+1=2\left(x-\left(\frac{\sqrt{2}}{2}-1\right)\right)\left(x-\left(-\frac{\sqrt{2}}{2}-1\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1+\frac{\sqrt{2}}{2} for x_{1} and -1-\frac{\sqrt{2}}{2} for x_{2}.
x ^ 2 +2x +\frac{1}{2} = 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.This is achieved by dividing both sides of the equation by 2
r + s = -2 rs = \frac{1}{2}
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 = -1 - u s = -1 + u
Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. 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>
(-1 - u) (-1 + u) = \frac{1}{2}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{2}
1 - u^2 = \frac{1}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{1}{2}-1 = -\frac{1}{2}
Simplify the expression by subtracting 1 on both sides
u^2 = \frac{1}{2} u = \pm\sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-1 - \frac{1}{\sqrt{2}} = -1.707 s = -1 + \frac{1}{\sqrt{2}} = -0.293
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.