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x ^ 2 +\frac{2}{3}x +3 = 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 3
r + s = -\frac{2}{3} rs = 3
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 = -\frac{1}{3} - u s = -\frac{1}{3} + u
Two numbers r and s sum up to -\frac{2}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{2}{3} = -\frac{1}{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>
(-\frac{1}{3} - u) (-\frac{1}{3} + u) = 3
To solve for unknown quantity u, substitute these in the product equation rs = 3
\frac{1}{9} - u^2 = 3
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 3-\frac{1}{9} = \frac{26}{9}
Simplify the expression by subtracting \frac{1}{9} on both sides
u^2 = -\frac{26}{9} u = \pm\sqrt{-\frac{26}{9}} = \pm \frac{\sqrt{26}}{3}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{1}{3} - \frac{\sqrt{26}}{3}i = -0.333 - 1.700i s = -\frac{1}{3} + \frac{\sqrt{26}}{3}i = -0.333 + 1.700i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.