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x\left(x+1\right)+36x\times \frac{1}{9}=36
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 36x, the least common multiple of 36,9,x.
x^{2}+x+36x\times \frac{1}{9}=36
Use the distributive property to multiply x by x+1.
x^{2}+x+\frac{36}{9}x=36
Multiply 36 and \frac{1}{9} to get \frac{36}{9}.
x^{2}+x+4x=36
Divide 36 by 9 to get 4.
x^{2}+5x=36
Combine x and 4x to get 5x.
x^{2}+5x-36=0
Subtract 36 from both sides.
x=\frac{-5±\sqrt{5^{2}-4\left(-36\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-5±\sqrt{25-4\left(-36\right)}}{2}
Square 5.
x=\frac{-5±\sqrt{25+144}}{2}
Multiply -4 times -36.
x=\frac{-5±\sqrt{169}}{2}
Add 25 to 144.
x=\frac{-5±13}{2}
Take the square root of 169.
x=\frac{8}{2}
Now solve the equation x=\frac{-5±13}{2} when ± is plus. Add -5 to 13.
x=4
Divide 8 by 2.
x=-\frac{18}{2}
Now solve the equation x=\frac{-5±13}{2} when ± is minus. Subtract 13 from -5.
x=-9
Divide -18 by 2.
x=4 x=-9
The equation is now solved.
x\left(x+1\right)+36x\times \frac{1}{9}=36
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 36x, the least common multiple of 36,9,x.
x^{2}+x+36x\times \frac{1}{9}=36
Use the distributive property to multiply x by x+1.
x^{2}+x+\frac{36}{9}x=36
Multiply 36 and \frac{1}{9} to get \frac{36}{9}.
x^{2}+x+4x=36
Divide 36 by 9 to get 4.
x^{2}+5x=36
Combine x and 4x to get 5x.
x^{2}+5x+\left(\frac{5}{2}\right)^{2}=36+\left(\frac{5}{2}\right)^{2}
Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+5x+\frac{25}{4}=36+\frac{25}{4}
Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+5x+\frac{25}{4}=\frac{169}{4}
Add 36 to \frac{25}{4}.
\left(x+\frac{5}{2}\right)^{2}=\frac{169}{4}
Factor x^{2}+5x+\frac{25}{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{5}{2}\right)^{2}}=\sqrt{\frac{169}{4}}
Take the square root of both sides of the equation.
x+\frac{5}{2}=\frac{13}{2} x+\frac{5}{2}=-\frac{13}{2}
Simplify.
x=4 x=-9
Subtract \frac{5}{2} from both sides of the equation.