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532\times 2=x\left(x-1\right)\times 3
Multiply both sides by 2.
1064=x\left(x-1\right)\times 3
Multiply 532 and 2 to get 1064.
1064=\left(x^{2}-x\right)\times 3
Use the distributive property to multiply x by x-1.
1064=3x^{2}-3x
Use the distributive property to multiply x^{2}-x by 3.
3x^{2}-3x=1064
Swap sides so that all variable terms are on the left hand side.
3x^{2}-3x-1064=0
Subtract 1064 from both sides.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 3\left(-1064\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -3 for b, and -1064 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 3\left(-1064\right)}}{2\times 3}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-12\left(-1064\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-3\right)±\sqrt{9+12768}}{2\times 3}
Multiply -12 times -1064.
x=\frac{-\left(-3\right)±\sqrt{12777}}{2\times 3}
Add 9 to 12768.
x=\frac{3±\sqrt{12777}}{2\times 3}
The opposite of -3 is 3.
x=\frac{3±\sqrt{12777}}{6}
Multiply 2 times 3.
x=\frac{\sqrt{12777}+3}{6}
Now solve the equation x=\frac{3±\sqrt{12777}}{6} when ± is plus. Add 3 to \sqrt{12777}.
x=\frac{\sqrt{12777}}{6}+\frac{1}{2}
Divide 3+\sqrt{12777} by 6.
x=\frac{3-\sqrt{12777}}{6}
Now solve the equation x=\frac{3±\sqrt{12777}}{6} when ± is minus. Subtract \sqrt{12777} from 3.
x=-\frac{\sqrt{12777}}{6}+\frac{1}{2}
Divide 3-\sqrt{12777} by 6.
x=\frac{\sqrt{12777}}{6}+\frac{1}{2} x=-\frac{\sqrt{12777}}{6}+\frac{1}{2}
The equation is now solved.
532\times 2=x\left(x-1\right)\times 3
Multiply both sides by 2.
1064=x\left(x-1\right)\times 3
Multiply 532 and 2 to get 1064.
1064=\left(x^{2}-x\right)\times 3
Use the distributive property to multiply x by x-1.
1064=3x^{2}-3x
Use the distributive property to multiply x^{2}-x by 3.
3x^{2}-3x=1064
Swap sides so that all variable terms are on the left hand side.
\frac{3x^{2}-3x}{3}=\frac{1064}{3}
Divide both sides by 3.
x^{2}+\left(-\frac{3}{3}\right)x=\frac{1064}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-x=\frac{1064}{3}
Divide -3 by 3.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{1064}{3}+\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{1064}{3}+\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{4259}{12}
Add \frac{1064}{3} 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{4259}{12}
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{4259}{12}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{12777}}{6} x-\frac{1}{2}=-\frac{\sqrt{12777}}{6}
Simplify.
x=\frac{\sqrt{12777}}{6}+\frac{1}{2} x=-\frac{\sqrt{12777}}{6}+\frac{1}{2}
Add \frac{1}{2} to both sides of the equation.