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2\left(x+2\right)\times 2-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+2\right), the least common multiple of x+2,2.
\left(2x+4\right)\times 2-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Use the distributive property to multiply 2 by x+2.
4x+8-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Use the distributive property to multiply 2x+4 by 2.
4x+8-2x+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Use the distributive property to multiply -2 by x-2.
2x+8+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Combine 4x and -2x to get 2x.
2x+12=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Add 8 and 4 to get 12.
2x+12=3\left(-x+2\right)\left(x+2\right)
Multiply \frac{3}{2} and 2 to get 3.
2x+12=\left(3\left(-x\right)+6\right)\left(x+2\right)
Use the distributive property to multiply 3 by -x+2.
2x+12=3\left(-x\right)x+6\left(-x\right)+6x+12
Use the distributive property to multiply 3\left(-x\right)+6 by x+2.
2x+12-3\left(-x\right)x=6\left(-x\right)+6x+12
Subtract 3\left(-x\right)x from both sides.
2x+12-3\left(-x\right)x-6\left(-x\right)=6x+12
Subtract 6\left(-x\right) from both sides.
2x+12-3\left(-x\right)x-6\left(-x\right)-6x=12
Subtract 6x from both sides.
2x+12-3\left(-x\right)x-6\left(-x\right)-6x-12=0
Subtract 12 from both sides.
2x+12-3\left(-1\right)x^{2}-6\left(-1\right)x-6x-12=0
Multiply x and x to get x^{2}.
2x+12+3x^{2}-6\left(-1\right)x-6x-12=0
Multiply -3 and -1 to get 3.
2x+12+3x^{2}+6x-6x-12=0
Multiply -6 and -1 to get 6.
8x+12+3x^{2}-6x-12=0
Combine 2x and 6x to get 8x.
2x+12+3x^{2}-12=0
Combine 8x and -6x to get 2x.
2x+3x^{2}=0
Subtract 12 from 12 to get 0.
3x^{2}+2x=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{-2±\sqrt{2^{2}}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±2}{2\times 3}
Take the square root of 2^{2}.
x=\frac{-2±2}{6}
Multiply 2 times 3.
x=\frac{0}{6}
Now solve the equation x=\frac{-2±2}{6} when ± is plus. Add -2 to 2.
x=0
Divide 0 by 6.
x=-\frac{4}{6}
Now solve the equation x=\frac{-2±2}{6} when ± is minus. Subtract 2 from -2.
x=-\frac{2}{3}
Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.
x=0 x=-\frac{2}{3}
The equation is now solved.
2\left(x+2\right)\times 2-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+2\right), the least common multiple of x+2,2.
\left(2x+4\right)\times 2-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Use the distributive property to multiply 2 by x+2.
4x+8-2\left(x-2\right)=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Use the distributive property to multiply 2x+4 by 2.
4x+8-2x+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Use the distributive property to multiply -2 by x-2.
2x+8+4=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Combine 4x and -2x to get 2x.
2x+12=\frac{3}{2}\left(-x+2\right)\times 2\left(x+2\right)
Add 8 and 4 to get 12.
2x+12=3\left(-x+2\right)\left(x+2\right)
Multiply \frac{3}{2} and 2 to get 3.
2x+12=\left(3\left(-x\right)+6\right)\left(x+2\right)
Use the distributive property to multiply 3 by -x+2.
2x+12=3\left(-x\right)x+6\left(-x\right)+6x+12
Use the distributive property to multiply 3\left(-x\right)+6 by x+2.
2x+12-3\left(-x\right)x=6\left(-x\right)+6x+12
Subtract 3\left(-x\right)x from both sides.
2x+12-3\left(-x\right)x-6\left(-x\right)=6x+12
Subtract 6\left(-x\right) from both sides.
2x+12-3\left(-x\right)x-6\left(-x\right)-6x=12
Subtract 6x from both sides.
2x+12-3\left(-1\right)x^{2}-6\left(-1\right)x-6x=12
Multiply x and x to get x^{2}.
2x+12+3x^{2}-6\left(-1\right)x-6x=12
Multiply -3 and -1 to get 3.
2x+12+3x^{2}+6x-6x=12
Multiply -6 and -1 to get 6.
8x+12+3x^{2}-6x=12
Combine 2x and 6x to get 8x.
2x+12+3x^{2}=12
Combine 8x and -6x to get 2x.
2x+3x^{2}=12-12
Subtract 12 from both sides.
2x+3x^{2}=0
Subtract 12 from 12 to get 0.
3x^{2}+2x=0
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{3x^{2}+2x}{3}=\frac{0}{3}
Divide both sides by 3.
x^{2}+\frac{2}{3}x=\frac{0}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{2}{3}x=0
Divide 0 by 3.
x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=\left(\frac{1}{3}\right)^{2}
Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{1}{9}
Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.
\left(x+\frac{1}{3}\right)^{2}=\frac{1}{9}
Factor x^{2}+\frac{2}{3}x+\frac{1}{9}. 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{1}{3}\right)^{2}}=\sqrt{\frac{1}{9}}
Take the square root of both sides of the equation.
x+\frac{1}{3}=\frac{1}{3} x+\frac{1}{3}=-\frac{1}{3}
Simplify.
x=0 x=-\frac{2}{3}
Subtract \frac{1}{3} from both sides of the equation.