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9x^{2}-22x+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(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 9\times 4}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -22 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-22\right)±\sqrt{484-4\times 9\times 4}}{2\times 9}
Square -22.
x=\frac{-\left(-22\right)±\sqrt{484-36\times 4}}{2\times 9}
Multiply -4 times 9.
x=\frac{-\left(-22\right)±\sqrt{484-144}}{2\times 9}
Multiply -36 times 4.
x=\frac{-\left(-22\right)±\sqrt{340}}{2\times 9}
Add 484 to -144.
x=\frac{-\left(-22\right)±2\sqrt{85}}{2\times 9}
Take the square root of 340.
x=\frac{22±2\sqrt{85}}{2\times 9}
The opposite of -22 is 22.
x=\frac{22±2\sqrt{85}}{18}
Multiply 2 times 9.
x=\frac{2\sqrt{85}+22}{18}
Now solve the equation x=\frac{22±2\sqrt{85}}{18} when ± is plus. Add 22 to 2\sqrt{85}.
x=\frac{\sqrt{85}+11}{9}
Divide 22+2\sqrt{85} by 18.
x=\frac{22-2\sqrt{85}}{18}
Now solve the equation x=\frac{22±2\sqrt{85}}{18} when ± is minus. Subtract 2\sqrt{85} from 22.
x=\frac{11-\sqrt{85}}{9}
Divide 22-2\sqrt{85} by 18.
x=\frac{\sqrt{85}+11}{9} x=\frac{11-\sqrt{85}}{9}
The equation is now solved.
9x^{2}-22x+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.
9x^{2}-22x+4-4=-4
Subtract 4 from both sides of the equation.
9x^{2}-22x=-4
Subtracting 4 from itself leaves 0.
\frac{9x^{2}-22x}{9}=-\frac{4}{9}
Divide both sides by 9.
x^{2}-\frac{22}{9}x=-\frac{4}{9}
Dividing by 9 undoes the multiplication by 9.
x^{2}-\frac{22}{9}x+\left(-\frac{11}{9}\right)^{2}=-\frac{4}{9}+\left(-\frac{11}{9}\right)^{2}
Divide -\frac{22}{9}, the coefficient of the x term, by 2 to get -\frac{11}{9}. Then add the square of -\frac{11}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{22}{9}x+\frac{121}{81}=-\frac{4}{9}+\frac{121}{81}
Square -\frac{11}{9} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{22}{9}x+\frac{121}{81}=\frac{85}{81}
Add -\frac{4}{9} to \frac{121}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{11}{9}\right)^{2}=\frac{85}{81}
Factor x^{2}-\frac{22}{9}x+\frac{121}{81}. 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{11}{9}\right)^{2}}=\sqrt{\frac{85}{81}}
Take the square root of both sides of the equation.
x-\frac{11}{9}=\frac{\sqrt{85}}{9} x-\frac{11}{9}=-\frac{\sqrt{85}}{9}
Simplify.
x=\frac{\sqrt{85}+11}{9} x=\frac{11-\sqrt{85}}{9}
Add \frac{11}{9} to both sides of the equation.
x ^ 2 -\frac{22}{9}x +\frac{4}{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 9
r + s = \frac{22}{9} rs = \frac{4}{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{11}{9} - u s = \frac{11}{9} + u
Two numbers r and s sum up to \frac{22}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{22}{9} = \frac{11}{9}. 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{11}{9} - u) (\frac{11}{9} + u) = \frac{4}{9}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{9}
\frac{121}{81} - u^2 = \frac{4}{9}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{4}{9}-\frac{121}{81} = -\frac{85}{81}
Simplify the expression by subtracting \frac{121}{81} on both sides
u^2 = \frac{85}{81} u = \pm\sqrt{\frac{85}{81}} = \pm \frac{\sqrt{85}}{9}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{11}{9} - \frac{\sqrt{85}}{9} = 0.198 s = \frac{11}{9} + \frac{\sqrt{85}}{9} = 2.247
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.