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-4\times 2x^{2}+18\times 18=9\left(x+1\right)
Multiply both sides of the equation by 36, the least common multiple of 9,2,4.
-8x^{2}+18\times 18=9\left(x+1\right)
Multiply -4 and 2 to get -8.
-8x^{2}+324=9\left(x+1\right)
Multiply 18 and 18 to get 324.
-8x^{2}+324=9x+9
Use the distributive property to multiply 9 by x+1.
-8x^{2}+324-9x=9
Subtract 9x from both sides.
-8x^{2}+324-9x-9=0
Subtract 9 from both sides.
-8x^{2}+315-9x=0
Subtract 9 from 324 to get 315.
-8x^{2}-9x+315=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(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-8\right)\times 315}}{2\left(-8\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -9 for b, and 315 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\left(-8\right)\times 315}}{2\left(-8\right)}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81+32\times 315}}{2\left(-8\right)}
Multiply -4 times -8.
x=\frac{-\left(-9\right)±\sqrt{81+10080}}{2\left(-8\right)}
Multiply 32 times 315.
x=\frac{-\left(-9\right)±\sqrt{10161}}{2\left(-8\right)}
Add 81 to 10080.
x=\frac{-\left(-9\right)±3\sqrt{1129}}{2\left(-8\right)}
Take the square root of 10161.
x=\frac{9±3\sqrt{1129}}{2\left(-8\right)}
The opposite of -9 is 9.
x=\frac{9±3\sqrt{1129}}{-16}
Multiply 2 times -8.
x=\frac{3\sqrt{1129}+9}{-16}
Now solve the equation x=\frac{9±3\sqrt{1129}}{-16} when ± is plus. Add 9 to 3\sqrt{1129}.
x=\frac{-3\sqrt{1129}-9}{16}
Divide 9+3\sqrt{1129} by -16.
x=\frac{9-3\sqrt{1129}}{-16}
Now solve the equation x=\frac{9±3\sqrt{1129}}{-16} when ± is minus. Subtract 3\sqrt{1129} from 9.
x=\frac{3\sqrt{1129}-9}{16}
Divide 9-3\sqrt{1129} by -16.
x=\frac{-3\sqrt{1129}-9}{16} x=\frac{3\sqrt{1129}-9}{16}
The equation is now solved.
-4\times 2x^{2}+18\times 18=9\left(x+1\right)
Multiply both sides of the equation by 36, the least common multiple of 9,2,4.
-8x^{2}+18\times 18=9\left(x+1\right)
Multiply -4 and 2 to get -8.
-8x^{2}+324=9\left(x+1\right)
Multiply 18 and 18 to get 324.
-8x^{2}+324=9x+9
Use the distributive property to multiply 9 by x+1.
-8x^{2}+324-9x=9
Subtract 9x from both sides.
-8x^{2}-9x=9-324
Subtract 324 from both sides.
-8x^{2}-9x=-315
Subtract 324 from 9 to get -315.
\frac{-8x^{2}-9x}{-8}=-\frac{315}{-8}
Divide both sides by -8.
x^{2}+\left(-\frac{9}{-8}\right)x=-\frac{315}{-8}
Dividing by -8 undoes the multiplication by -8.
x^{2}+\frac{9}{8}x=-\frac{315}{-8}
Divide -9 by -8.
x^{2}+\frac{9}{8}x=\frac{315}{8}
Divide -315 by -8.
x^{2}+\frac{9}{8}x+\left(\frac{9}{16}\right)^{2}=\frac{315}{8}+\left(\frac{9}{16}\right)^{2}
Divide \frac{9}{8}, the coefficient of the x term, by 2 to get \frac{9}{16}. Then add the square of \frac{9}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{9}{8}x+\frac{81}{256}=\frac{315}{8}+\frac{81}{256}
Square \frac{9}{16} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{9}{8}x+\frac{81}{256}=\frac{10161}{256}
Add \frac{315}{8} to \frac{81}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{9}{16}\right)^{2}=\frac{10161}{256}
Factor x^{2}+\frac{9}{8}x+\frac{81}{256}. 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{9}{16}\right)^{2}}=\sqrt{\frac{10161}{256}}
Take the square root of both sides of the equation.
x+\frac{9}{16}=\frac{3\sqrt{1129}}{16} x+\frac{9}{16}=-\frac{3\sqrt{1129}}{16}
Simplify.
x=\frac{3\sqrt{1129}-9}{16} x=\frac{-3\sqrt{1129}-9}{16}
Subtract \frac{9}{16} from both sides of the equation.