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4x+4x+8=3x\left(x+2\right)
Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+2\right), the least common multiple of x+2,x,4.
8x+8=3x\left(x+2\right)
Combine 4x and 4x to get 8x.
8x+8=3x^{2}+6x
Use the distributive property to multiply 3x by x+2.
8x+8-3x^{2}=6x
Subtract 3x^{2} from both sides.
8x+8-3x^{2}-6x=0
Subtract 6x from both sides.
2x+8-3x^{2}=0
Combine 8x and -6x to get 2x.
-3x^{2}+2x+8=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=2 ab=-3\times 8=-24
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+8. To find a and b, set up a system to be solved.
-1,24 -2,12 -3,8 -4,6
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.
-1+24=23 -2+12=10 -3+8=5 -4+6=2
Calculate the sum for each pair.
a=6 b=-4
The solution is the pair that gives sum 2.
\left(-3x^{2}+6x\right)+\left(-4x+8\right)
Rewrite -3x^{2}+2x+8 as \left(-3x^{2}+6x\right)+\left(-4x+8\right).
3x\left(-x+2\right)+4\left(-x+2\right)
Factor out 3x in the first and 4 in the second group.
\left(-x+2\right)\left(3x+4\right)
Factor out common term -x+2 by using distributive property.
x=2 x=-\frac{4}{3}
To find equation solutions, solve -x+2=0 and 3x+4=0.
4x+4x+8=3x\left(x+2\right)
Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+2\right), the least common multiple of x+2,x,4.
8x+8=3x\left(x+2\right)
Combine 4x and 4x to get 8x.
8x+8=3x^{2}+6x
Use the distributive property to multiply 3x by x+2.
8x+8-3x^{2}=6x
Subtract 3x^{2} from both sides.
8x+8-3x^{2}-6x=0
Subtract 6x from both sides.
2x+8-3x^{2}=0
Combine 8x and -6x to get 2x.
-3x^{2}+2x+8=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}-4\left(-3\right)\times 8}}{2\left(-3\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 2 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-3\right)\times 8}}{2\left(-3\right)}
Square 2.
x=\frac{-2±\sqrt{4+12\times 8}}{2\left(-3\right)}
Multiply -4 times -3.
x=\frac{-2±\sqrt{4+96}}{2\left(-3\right)}
Multiply 12 times 8.
x=\frac{-2±\sqrt{100}}{2\left(-3\right)}
Add 4 to 96.
x=\frac{-2±10}{2\left(-3\right)}
Take the square root of 100.
x=\frac{-2±10}{-6}
Multiply 2 times -3.
x=\frac{8}{-6}
Now solve the equation x=\frac{-2±10}{-6} when ± is plus. Add -2 to 10.
x=-\frac{4}{3}
Reduce the fraction \frac{8}{-6} to lowest terms by extracting and canceling out 2.
x=-\frac{12}{-6}
Now solve the equation x=\frac{-2±10}{-6} when ± is minus. Subtract 10 from -2.
x=2
Divide -12 by -6.
x=-\frac{4}{3} x=2
The equation is now solved.
4x+4x+8=3x\left(x+2\right)
Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+2\right), the least common multiple of x+2,x,4.
8x+8=3x\left(x+2\right)
Combine 4x and 4x to get 8x.
8x+8=3x^{2}+6x
Use the distributive property to multiply 3x by x+2.
8x+8-3x^{2}=6x
Subtract 3x^{2} from both sides.
8x+8-3x^{2}-6x=0
Subtract 6x from both sides.
2x+8-3x^{2}=0
Combine 8x and -6x to get 2x.
2x-3x^{2}=-8
Subtract 8 from both sides. Anything subtracted from zero gives its negation.
-3x^{2}+2x=-8
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{8}{-3}
Divide both sides by -3.
x^{2}+\frac{2}{-3}x=-\frac{8}{-3}
Dividing by -3 undoes the multiplication by -3.
x^{2}-\frac{2}{3}x=-\frac{8}{-3}
Divide 2 by -3.
x^{2}-\frac{2}{3}x=\frac{8}{3}
Divide -8 by -3.
x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=\frac{8}{3}+\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{8}{3}+\frac{1}{9}
Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{25}{9}
Add \frac{8}{3} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{3}\right)^{2}=\frac{25}{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{25}{9}}
Take the square root of both sides of the equation.
x-\frac{1}{3}=\frac{5}{3} x-\frac{1}{3}=-\frac{5}{3}
Simplify.
x=2 x=-\frac{4}{3}
Add \frac{1}{3} to both sides of the equation.