Solve for x (complex solution)
x=\frac{-\sqrt{151}i-3}{8}\approx -0.375-1.536025716i
x=\frac{-3+\sqrt{151}i}{8}\approx -0.375+1.536025716i
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\left(3x+2\right)\left(x-11\right)=\left(3x-2\right)\left(5x-4\right)
Variable x cannot be equal to any of the values -\frac{2}{3},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-2\right)\left(3x+2\right), the least common multiple of 3x-2,3x+2.
3x^{2}-31x-22=\left(3x-2\right)\left(5x-4\right)
Use the distributive property to multiply 3x+2 by x-11 and combine like terms.
3x^{2}-31x-22=15x^{2}-22x+8
Use the distributive property to multiply 3x-2 by 5x-4 and combine like terms.
3x^{2}-31x-22-15x^{2}=-22x+8
Subtract 15x^{2} from both sides.
-12x^{2}-31x-22=-22x+8
Combine 3x^{2} and -15x^{2} to get -12x^{2}.
-12x^{2}-31x-22+22x=8
Add 22x to both sides.
-12x^{2}-9x-22=8
Combine -31x and 22x to get -9x.
-12x^{2}-9x-22-8=0
Subtract 8 from both sides.
-12x^{2}-9x-30=0
Subtract 8 from -22 to get -30.
x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-12\right)\left(-30\right)}}{2\left(-12\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, -9 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\left(-12\right)\left(-30\right)}}{2\left(-12\right)}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81+48\left(-30\right)}}{2\left(-12\right)}
Multiply -4 times -12.
x=\frac{-\left(-9\right)±\sqrt{81-1440}}{2\left(-12\right)}
Multiply 48 times -30.
x=\frac{-\left(-9\right)±\sqrt{-1359}}{2\left(-12\right)}
Add 81 to -1440.
x=\frac{-\left(-9\right)±3\sqrt{151}i}{2\left(-12\right)}
Take the square root of -1359.
x=\frac{9±3\sqrt{151}i}{2\left(-12\right)}
The opposite of -9 is 9.
x=\frac{9±3\sqrt{151}i}{-24}
Multiply 2 times -12.
x=\frac{9+3\sqrt{151}i}{-24}
Now solve the equation x=\frac{9±3\sqrt{151}i}{-24} when ± is plus. Add 9 to 3i\sqrt{151}.
x=\frac{-\sqrt{151}i-3}{8}
Divide 9+3i\sqrt{151} by -24.
x=\frac{-3\sqrt{151}i+9}{-24}
Now solve the equation x=\frac{9±3\sqrt{151}i}{-24} when ± is minus. Subtract 3i\sqrt{151} from 9.
x=\frac{-3+\sqrt{151}i}{8}
Divide 9-3i\sqrt{151} by -24.
x=\frac{-\sqrt{151}i-3}{8} x=\frac{-3+\sqrt{151}i}{8}
The equation is now solved.
\left(3x+2\right)\left(x-11\right)=\left(3x-2\right)\left(5x-4\right)
Variable x cannot be equal to any of the values -\frac{2}{3},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-2\right)\left(3x+2\right), the least common multiple of 3x-2,3x+2.
3x^{2}-31x-22=\left(3x-2\right)\left(5x-4\right)
Use the distributive property to multiply 3x+2 by x-11 and combine like terms.
3x^{2}-31x-22=15x^{2}-22x+8
Use the distributive property to multiply 3x-2 by 5x-4 and combine like terms.
3x^{2}-31x-22-15x^{2}=-22x+8
Subtract 15x^{2} from both sides.
-12x^{2}-31x-22=-22x+8
Combine 3x^{2} and -15x^{2} to get -12x^{2}.
-12x^{2}-31x-22+22x=8
Add 22x to both sides.
-12x^{2}-9x-22=8
Combine -31x and 22x to get -9x.
-12x^{2}-9x=8+22
Add 22 to both sides.
-12x^{2}-9x=30
Add 8 and 22 to get 30.
\frac{-12x^{2}-9x}{-12}=\frac{30}{-12}
Divide both sides by -12.
x^{2}+\left(-\frac{9}{-12}\right)x=\frac{30}{-12}
Dividing by -12 undoes the multiplication by -12.
x^{2}+\frac{3}{4}x=\frac{30}{-12}
Reduce the fraction \frac{-9}{-12} to lowest terms by extracting and canceling out 3.
x^{2}+\frac{3}{4}x=-\frac{5}{2}
Reduce the fraction \frac{30}{-12} to lowest terms by extracting and canceling out 6.
x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{5}{2}+\left(\frac{3}{8}\right)^{2}
Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{5}{2}+\frac{9}{64}
Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{151}{64}
Add -\frac{5}{2} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{3}{8}\right)^{2}=-\frac{151}{64}
Factor x^{2}+\frac{3}{4}x+\frac{9}{64}. 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{3}{8}\right)^{2}}=\sqrt{-\frac{151}{64}}
Take the square root of both sides of the equation.
x+\frac{3}{8}=\frac{\sqrt{151}i}{8} x+\frac{3}{8}=-\frac{\sqrt{151}i}{8}
Simplify.
x=\frac{-3+\sqrt{151}i}{8} x=\frac{-\sqrt{151}i-3}{8}
Subtract \frac{3}{8} from both sides of the equation.
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