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6x^{2}-8x+3=0
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{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 6\times 3}}{2\times 6}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -8 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-8\right)±\sqrt{64-4\times 6\times 3}}{2\times 6}
Square -8.
x=\frac{-\left(-8\right)±\sqrt{64-24\times 3}}{2\times 6}
Multiply -4 times 6.
x=\frac{-\left(-8\right)±\sqrt{64-72}}{2\times 6}
Multiply -24 times 3.
x=\frac{-\left(-8\right)±\sqrt{-8}}{2\times 6}
Add 64 to -72.
x=\frac{-\left(-8\right)±2\sqrt{2}i}{2\times 6}
Take the square root of -8.
x=\frac{8±2\sqrt{2}i}{2\times 6}
The opposite of -8 is 8.
x=\frac{8±2\sqrt{2}i}{12}
Multiply 2 times 6.
x=\frac{8+2\sqrt{2}i}{12}
Now solve the equation x=\frac{8±2\sqrt{2}i}{12} when ± is plus. Add 8 to 2i\sqrt{2}.
x=\frac{\sqrt{2}i}{6}+\frac{2}{3}
Divide 8+2i\sqrt{2} by 12.
x=\frac{-2\sqrt{2}i+8}{12}
Now solve the equation x=\frac{8±2\sqrt{2}i}{12} when ± is minus. Subtract 2i\sqrt{2} from 8.
x=-\frac{\sqrt{2}i}{6}+\frac{2}{3}
Divide 8-2i\sqrt{2} by 12.
x=\frac{\sqrt{2}i}{6}+\frac{2}{3} x=-\frac{\sqrt{2}i}{6}+\frac{2}{3}
The equation is now solved.
6x^{2}-8x+3=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
6x^{2}-8x+3-3=-3
Subtract 3 from both sides of the equation.
6x^{2}-8x=-3
Subtracting 3 from itself leaves 0.
\frac{6x^{2}-8x}{6}=-\frac{3}{6}
Divide both sides by 6.
x^{2}+\left(-\frac{8}{6}\right)x=-\frac{3}{6}
Dividing by 6 undoes the multiplication by 6.
x^{2}-\frac{4}{3}x=-\frac{3}{6}
Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{4}{3}x=-\frac{1}{2}
Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.
x^{2}-\frac{4}{3}x+\left(-\frac{2}{3}\right)^{2}=-\frac{1}{2}+\left(-\frac{2}{3}\right)^{2}
Divide -\frac{4}{3}, the coefficient of the x term, by 2 to get -\frac{2}{3}. Then add the square of -\frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{4}{3}x+\frac{4}{9}=-\frac{1}{2}+\frac{4}{9}
Square -\frac{2}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{4}{3}x+\frac{4}{9}=-\frac{1}{18}
Add -\frac{1}{2} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{2}{3}\right)^{2}=-\frac{1}{18}
Factor x^{2}-\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{-\frac{1}{18}}
Take the square root of both sides of the equation.
x-\frac{2}{3}=\frac{\sqrt{2}i}{6} x-\frac{2}{3}=-\frac{\sqrt{2}i}{6}
Simplify.
x=\frac{\sqrt{2}i}{6}+\frac{2}{3} x=-\frac{\sqrt{2}i}{6}+\frac{2}{3}
Add \frac{2}{3} to both sides of the equation.
x ^ 2 -\frac{4}{3}x +\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 6
r + s = \frac{4}{3} 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 = \frac{2}{3} - u s = \frac{2}{3} + u
Two numbers r and s sum up to \frac{4}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{4}{3} = \frac{2}{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{2}{3} - u) (\frac{2}{3} + u) = \frac{1}{2}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{2}
\frac{4}{9} - u^2 = \frac{1}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{1}{2}-\frac{4}{9} = \frac{1}{18}
Simplify the expression by subtracting \frac{4}{9} on both sides
u^2 = -\frac{1}{18} u = \pm\sqrt{-\frac{1}{18}} = \pm \frac{1}{\sqrt{18}}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{2}{3} - \frac{1}{\sqrt{18}}i = 0.667 - 0.236i s = \frac{2}{3} + \frac{1}{\sqrt{18}}i = 0.667 + 0.236i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.