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Solve for x (complex solution)
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\left(6-2x\right)^{2}=-2x\left(x+3\right)
Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right).
36-24x+4x^{2}=-2x\left(x+3\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-2x\right)^{2}.
36-24x+4x^{2}=-2x^{2}-6x
Use the distributive property to multiply -2x by x+3.
36-24x+4x^{2}+2x^{2}=-6x
Add 2x^{2} to both sides.
36-24x+6x^{2}=-6x
Combine 4x^{2} and 2x^{2} to get 6x^{2}.
36-24x+6x^{2}+6x=0
Add 6x to both sides.
36-18x+6x^{2}=0
Combine -24x and 6x to get -18x.
6x^{2}-18x+36=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(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 6\times 36}}{2\times 6}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -18 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-18\right)±\sqrt{324-4\times 6\times 36}}{2\times 6}
Square -18.
x=\frac{-\left(-18\right)±\sqrt{324-24\times 36}}{2\times 6}
Multiply -4 times 6.
x=\frac{-\left(-18\right)±\sqrt{324-864}}{2\times 6}
Multiply -24 times 36.
x=\frac{-\left(-18\right)±\sqrt{-540}}{2\times 6}
Add 324 to -864.
x=\frac{-\left(-18\right)±6\sqrt{15}i}{2\times 6}
Take the square root of -540.
x=\frac{18±6\sqrt{15}i}{2\times 6}
The opposite of -18 is 18.
x=\frac{18±6\sqrt{15}i}{12}
Multiply 2 times 6.
x=\frac{18+6\sqrt{15}i}{12}
Now solve the equation x=\frac{18±6\sqrt{15}i}{12} when ± is plus. Add 18 to 6i\sqrt{15}.
x=\frac{3+\sqrt{15}i}{2}
Divide 18+6i\sqrt{15} by 12.
x=\frac{-6\sqrt{15}i+18}{12}
Now solve the equation x=\frac{18±6\sqrt{15}i}{12} when ± is minus. Subtract 6i\sqrt{15} from 18.
x=\frac{-\sqrt{15}i+3}{2}
Divide 18-6i\sqrt{15} by 12.
x=\frac{3+\sqrt{15}i}{2} x=\frac{-\sqrt{15}i+3}{2}
The equation is now solved.
\left(6-2x\right)^{2}=-2x\left(x+3\right)
Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right).
36-24x+4x^{2}=-2x\left(x+3\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-2x\right)^{2}.
36-24x+4x^{2}=-2x^{2}-6x
Use the distributive property to multiply -2x by x+3.
36-24x+4x^{2}+2x^{2}=-6x
Add 2x^{2} to both sides.
36-24x+6x^{2}=-6x
Combine 4x^{2} and 2x^{2} to get 6x^{2}.
36-24x+6x^{2}+6x=0
Add 6x to both sides.
36-18x+6x^{2}=0
Combine -24x and 6x to get -18x.
-18x+6x^{2}=-36
Subtract 36 from both sides. Anything subtracted from zero gives its negation.
6x^{2}-18x=-36
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{6x^{2}-18x}{6}=-\frac{36}{6}
Divide both sides by 6.
x^{2}+\left(-\frac{18}{6}\right)x=-\frac{36}{6}
Dividing by 6 undoes the multiplication by 6.
x^{2}-3x=-\frac{36}{6}
Divide -18 by 6.
x^{2}-3x=-6
Divide -36 by 6.
x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-6+\left(-\frac{3}{2}\right)^{2}
Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-3x+\frac{9}{4}=-6+\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-3x+\frac{9}{4}=-\frac{15}{4}
Add -6 to \frac{9}{4}.
\left(x-\frac{3}{2}\right)^{2}=-\frac{15}{4}
Factor x^{2}-3x+\frac{9}{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{3}{2}\right)^{2}}=\sqrt{-\frac{15}{4}}
Take the square root of both sides of the equation.
x-\frac{3}{2}=\frac{\sqrt{15}i}{2} x-\frac{3}{2}=-\frac{\sqrt{15}i}{2}
Simplify.
x=\frac{3+\sqrt{15}i}{2} x=\frac{-\sqrt{15}i+3}{2}
Add \frac{3}{2} to both sides of the equation.