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18x^{2}-18x-4=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(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 18\left(-4\right)}}{2\times 18}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, -18 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-18\right)±\sqrt{324-4\times 18\left(-4\right)}}{2\times 18}
Square -18.
x=\frac{-\left(-18\right)±\sqrt{324-72\left(-4\right)}}{2\times 18}
Multiply -4 times 18.
x=\frac{-\left(-18\right)±\sqrt{324+288}}{2\times 18}
Multiply -72 times -4.
x=\frac{-\left(-18\right)±\sqrt{612}}{2\times 18}
Add 324 to 288.
x=\frac{-\left(-18\right)±6\sqrt{17}}{2\times 18}
Take the square root of 612.
x=\frac{18±6\sqrt{17}}{2\times 18}
The opposite of -18 is 18.
x=\frac{18±6\sqrt{17}}{36}
Multiply 2 times 18.
x=\frac{6\sqrt{17}+18}{36}
Now solve the equation x=\frac{18±6\sqrt{17}}{36} when ± is plus. Add 18 to 6\sqrt{17}.
x=\frac{\sqrt{17}}{6}+\frac{1}{2}
Divide 18+6\sqrt{17} by 36.
x=\frac{18-6\sqrt{17}}{36}
Now solve the equation x=\frac{18±6\sqrt{17}}{36} when ± is minus. Subtract 6\sqrt{17} from 18.
x=-\frac{\sqrt{17}}{6}+\frac{1}{2}
Divide 18-6\sqrt{17} by 36.
x=\frac{\sqrt{17}}{6}+\frac{1}{2} x=-\frac{\sqrt{17}}{6}+\frac{1}{2}
The equation is now solved.
18x^{2}-18x-4=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.
18x^{2}-18x-4-\left(-4\right)=-\left(-4\right)
Add 4 to both sides of the equation.
18x^{2}-18x=-\left(-4\right)
Subtracting -4 from itself leaves 0.
18x^{2}-18x=4
Subtract -4 from 0.
\frac{18x^{2}-18x}{18}=\frac{4}{18}
Divide both sides by 18.
x^{2}+\left(-\frac{18}{18}\right)x=\frac{4}{18}
Dividing by 18 undoes the multiplication by 18.
x^{2}-x=\frac{4}{18}
Divide -18 by 18.
x^{2}-x=\frac{2}{9}
Reduce the fraction \frac{4}{18} to lowest terms by extracting and canceling out 2.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{2}{9}+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=\frac{2}{9}+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{17}{36}
Add \frac{2}{9} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{2}\right)^{2}=\frac{17}{36}
Factor x^{2}-x+\frac{1}{4}. 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{1}{2}\right)^{2}}=\sqrt{\frac{17}{36}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{17}}{6} x-\frac{1}{2}=-\frac{\sqrt{17}}{6}
Simplify.
x=\frac{\sqrt{17}}{6}+\frac{1}{2} x=-\frac{\sqrt{17}}{6}+\frac{1}{2}
Add \frac{1}{2} to both sides of the equation.
x ^ 2 -1x -\frac{2}{9} = 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 18
r + s = 1 rs = -\frac{2}{9}
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}{2} - u s = \frac{1}{2} + u
Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{2}. 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}{2} - u) (\frac{1}{2} + u) = -\frac{2}{9}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{9}
\frac{1}{4} - u^2 = -\frac{2}{9}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{2}{9}-\frac{1}{4} = -\frac{17}{36}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{17}{36} u = \pm\sqrt{\frac{17}{36}} = \pm \frac{\sqrt{17}}{6}
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}{2} - \frac{\sqrt{17}}{6} = -0.187 s = \frac{1}{2} + \frac{\sqrt{17}}{6} = 1.187
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.