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\left(3x+3\right)\times 2+\left(3x+9\right)\times 2=8\left(x+1\right)\left(x+3\right)
Variable x cannot be equal to any of the values -3,-1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(x+3\right), the least common multiple of x+3,x+1,3.
6x+6+\left(3x+9\right)\times 2=8\left(x+1\right)\left(x+3\right)
Use the distributive property to multiply 3x+3 by 2.
6x+6+6x+18=8\left(x+1\right)\left(x+3\right)
Use the distributive property to multiply 3x+9 by 2.
12x+6+18=8\left(x+1\right)\left(x+3\right)
Combine 6x and 6x to get 12x.
12x+24=8\left(x+1\right)\left(x+3\right)
Add 6 and 18 to get 24.
12x+24=\left(8x+8\right)\left(x+3\right)
Use the distributive property to multiply 8 by x+1.
12x+24=8x^{2}+32x+24
Use the distributive property to multiply 8x+8 by x+3 and combine like terms.
12x+24-8x^{2}=32x+24
Subtract 8x^{2} from both sides.
12x+24-8x^{2}-32x=24
Subtract 32x from both sides.
-20x+24-8x^{2}=24
Combine 12x and -32x to get -20x.
-20x+24-8x^{2}-24=0
Subtract 24 from both sides.
-20x-8x^{2}=0
Subtract 24 from 24 to get 0.
-8x^{2}-20x=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(-20\right)±\sqrt{\left(-20\right)^{2}}}{2\left(-8\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -20 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-20\right)±20}{2\left(-8\right)}
Take the square root of \left(-20\right)^{2}.
x=\frac{20±20}{2\left(-8\right)}
The opposite of -20 is 20.
x=\frac{20±20}{-16}
Multiply 2 times -8.
x=\frac{40}{-16}
Now solve the equation x=\frac{20±20}{-16} when ± is plus. Add 20 to 20.
x=-\frac{5}{2}
Reduce the fraction \frac{40}{-16} to lowest terms by extracting and canceling out 8.
x=\frac{0}{-16}
Now solve the equation x=\frac{20±20}{-16} when ± is minus. Subtract 20 from 20.
x=0
Divide 0 by -16.
x=-\frac{5}{2} x=0
The equation is now solved.
\left(3x+3\right)\times 2+\left(3x+9\right)\times 2=8\left(x+1\right)\left(x+3\right)
Variable x cannot be equal to any of the values -3,-1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(x+3\right), the least common multiple of x+3,x+1,3.
6x+6+\left(3x+9\right)\times 2=8\left(x+1\right)\left(x+3\right)
Use the distributive property to multiply 3x+3 by 2.
6x+6+6x+18=8\left(x+1\right)\left(x+3\right)
Use the distributive property to multiply 3x+9 by 2.
12x+6+18=8\left(x+1\right)\left(x+3\right)
Combine 6x and 6x to get 12x.
12x+24=8\left(x+1\right)\left(x+3\right)
Add 6 and 18 to get 24.
12x+24=\left(8x+8\right)\left(x+3\right)
Use the distributive property to multiply 8 by x+1.
12x+24=8x^{2}+32x+24
Use the distributive property to multiply 8x+8 by x+3 and combine like terms.
12x+24-8x^{2}=32x+24
Subtract 8x^{2} from both sides.
12x+24-8x^{2}-32x=24
Subtract 32x from both sides.
-20x+24-8x^{2}=24
Combine 12x and -32x to get -20x.
-20x-8x^{2}=24-24
Subtract 24 from both sides.
-20x-8x^{2}=0
Subtract 24 from 24 to get 0.
-8x^{2}-20x=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{-8x^{2}-20x}{-8}=\frac{0}{-8}
Divide both sides by -8.
x^{2}+\left(-\frac{20}{-8}\right)x=\frac{0}{-8}
Dividing by -8 undoes the multiplication by -8.
x^{2}+\frac{5}{2}x=\frac{0}{-8}
Reduce the fraction \frac{-20}{-8} to lowest terms by extracting and canceling out 4.
x^{2}+\frac{5}{2}x=0
Divide 0 by -8.
x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=\left(\frac{5}{4}\right)^{2}
Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{25}{16}
Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.
\left(x+\frac{5}{4}\right)^{2}=\frac{25}{16}
Factor x^{2}+\frac{5}{2}x+\frac{25}{16}. 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{5}{4}\right)^{2}}=\sqrt{\frac{25}{16}}
Take the square root of both sides of the equation.
x+\frac{5}{4}=\frac{5}{4} x+\frac{5}{4}=-\frac{5}{4}
Simplify.
x=0 x=-\frac{5}{2}
Subtract \frac{5}{4} from both sides of the equation.