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6x+6x+30=x\left(x+5\right)
Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+5\right), the least common multiple of x+5,x,6.
12x+30=x\left(x+5\right)
Combine 6x and 6x to get 12x.
12x+30=x^{2}+5x
Use the distributive property to multiply x by x+5.
12x+30-x^{2}=5x
Subtract x^{2} from both sides.
12x+30-x^{2}-5x=0
Subtract 5x from both sides.
7x+30-x^{2}=0
Combine 12x and -5x to get 7x.
-x^{2}+7x+30=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=7 ab=-30=-30
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+30. To find a and b, set up a system to be solved.
-1,30 -2,15 -3,10 -5,6
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 -30.
-1+30=29 -2+15=13 -3+10=7 -5+6=1
Calculate the sum for each pair.
a=10 b=-3
The solution is the pair that gives sum 7.
\left(-x^{2}+10x\right)+\left(-3x+30\right)
Rewrite -x^{2}+7x+30 as \left(-x^{2}+10x\right)+\left(-3x+30\right).
-x\left(x-10\right)-3\left(x-10\right)
Factor out -x in the first and -3 in the second group.
\left(x-10\right)\left(-x-3\right)
Factor out common term x-10 by using distributive property.
x=10 x=-3
To find equation solutions, solve x-10=0 and -x-3=0.
6x+6x+30=x\left(x+5\right)
Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+5\right), the least common multiple of x+5,x,6.
12x+30=x\left(x+5\right)
Combine 6x and 6x to get 12x.
12x+30=x^{2}+5x
Use the distributive property to multiply x by x+5.
12x+30-x^{2}=5x
Subtract x^{2} from both sides.
12x+30-x^{2}-5x=0
Subtract 5x from both sides.
7x+30-x^{2}=0
Combine 12x and -5x to get 7x.
-x^{2}+7x+30=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{-7±\sqrt{7^{2}-4\left(-1\right)\times 30}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\left(-1\right)\times 30}}{2\left(-1\right)}
Square 7.
x=\frac{-7±\sqrt{49+4\times 30}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-7±\sqrt{49+120}}{2\left(-1\right)}
Multiply 4 times 30.
x=\frac{-7±\sqrt{169}}{2\left(-1\right)}
Add 49 to 120.
x=\frac{-7±13}{2\left(-1\right)}
Take the square root of 169.
x=\frac{-7±13}{-2}
Multiply 2 times -1.
x=\frac{6}{-2}
Now solve the equation x=\frac{-7±13}{-2} when ± is plus. Add -7 to 13.
x=-3
Divide 6 by -2.
x=-\frac{20}{-2}
Now solve the equation x=\frac{-7±13}{-2} when ± is minus. Subtract 13 from -7.
x=10
Divide -20 by -2.
x=-3 x=10
The equation is now solved.
6x+6x+30=x\left(x+5\right)
Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+5\right), the least common multiple of x+5,x,6.
12x+30=x\left(x+5\right)
Combine 6x and 6x to get 12x.
12x+30=x^{2}+5x
Use the distributive property to multiply x by x+5.
12x+30-x^{2}=5x
Subtract x^{2} from both sides.
12x+30-x^{2}-5x=0
Subtract 5x from both sides.
7x+30-x^{2}=0
Combine 12x and -5x to get 7x.
7x-x^{2}=-30
Subtract 30 from both sides. Anything subtracted from zero gives its negation.
-x^{2}+7x=-30
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{-x^{2}+7x}{-1}=-\frac{30}{-1}
Divide both sides by -1.
x^{2}+\frac{7}{-1}x=-\frac{30}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-7x=-\frac{30}{-1}
Divide 7 by -1.
x^{2}-7x=30
Divide -30 by -1.
x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=30+\left(-\frac{7}{2}\right)^{2}
Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-7x+\frac{49}{4}=30+\frac{49}{4}
Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-7x+\frac{49}{4}=\frac{169}{4}
Add 30 to \frac{49}{4}.
\left(x-\frac{7}{2}\right)^{2}=\frac{169}{4}
Factor x^{2}-7x+\frac{49}{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{7}{2}\right)^{2}}=\sqrt{\frac{169}{4}}
Take the square root of both sides of the equation.
x-\frac{7}{2}=\frac{13}{2} x-\frac{7}{2}=-\frac{13}{2}
Simplify.
x=10 x=-3
Add \frac{7}{2} to both sides of the equation.