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Solve for x (complex solution)
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4x\left(x+3\right)=3\left(5x-1\right)
Multiply both sides of the equation by 12, the least common multiple of 3,4.
4x^{2}+12x=3\left(5x-1\right)
Use the distributive property to multiply 4x by x+3.
4x^{2}+12x=15x-3
Use the distributive property to multiply 3 by 5x-1.
4x^{2}+12x-15x=-3
Subtract 15x from both sides.
4x^{2}-3x=-3
Combine 12x and -15x to get -3x.
4x^{2}-3x+3=0
Add 3 to both sides.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 4\times 3}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -3 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 4\times 3}}{2\times 4}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-16\times 3}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-3\right)±\sqrt{9-48}}{2\times 4}
Multiply -16 times 3.
x=\frac{-\left(-3\right)±\sqrt{-39}}{2\times 4}
Add 9 to -48.
x=\frac{-\left(-3\right)±\sqrt{39}i}{2\times 4}
Take the square root of -39.
x=\frac{3±\sqrt{39}i}{2\times 4}
The opposite of -3 is 3.
x=\frac{3±\sqrt{39}i}{8}
Multiply 2 times 4.
x=\frac{3+\sqrt{39}i}{8}
Now solve the equation x=\frac{3±\sqrt{39}i}{8} when ± is plus. Add 3 to i\sqrt{39}.
x=\frac{-\sqrt{39}i+3}{8}
Now solve the equation x=\frac{3±\sqrt{39}i}{8} when ± is minus. Subtract i\sqrt{39} from 3.
x=\frac{3+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+3}{8}
The equation is now solved.
4x\left(x+3\right)=3\left(5x-1\right)
Multiply both sides of the equation by 12, the least common multiple of 3,4.
4x^{2}+12x=3\left(5x-1\right)
Use the distributive property to multiply 4x by x+3.
4x^{2}+12x=15x-3
Use the distributive property to multiply 3 by 5x-1.
4x^{2}+12x-15x=-3
Subtract 15x from both sides.
4x^{2}-3x=-3
Combine 12x and -15x to get -3x.
\frac{4x^{2}-3x}{4}=-\frac{3}{4}
Divide both sides by 4.
x^{2}-\frac{3}{4}x=-\frac{3}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=-\frac{3}{4}+\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{3}{4}+\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{39}{64}
Add -\frac{3}{4} 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{39}{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{39}{64}}
Take the square root of both sides of the equation.
x-\frac{3}{8}=\frac{\sqrt{39}i}{8} x-\frac{3}{8}=-\frac{\sqrt{39}i}{8}
Simplify.
x=\frac{3+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+3}{8}
Add \frac{3}{8} to both sides of the equation.