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(3x(x+10)^(2))/(x^(2))=((x+10)^(3))/(x^(2))

Reduce the expression (3x(x+10)^(2))/(x^(2)) by removing a factor of x from the numerator and denominator.

(3(x+10)^(2))/(x)=((x+10)^(3))/(x^(2))

Since there is one rational expression on each side of the equation, this can be solved as a ratio. For example, (A)/(B)=(C)/(D) is equivalent to A*D=B*C.

3(x+10)^(2)*x^(2)=(x+10)^(3)*x

Multiply 3 by x^(2) to get 3x^(2).

3x^(2)(x+10)^(2)=(x+10)^(3)*x

Multiply (x+10)^(3) by x to get x(x+10)^(3).

3x^(2)(x+10)^(2)=x(x+10)^(3)

Since x(x+10)^(3) contains the variable to solve for, move it to the left-hand side of the equation by subtracting x(x+10)^(3) from both sides.

3x^(2)(x+10)^(2)-x(x+10)^(3)=0

Squaring an expression is the same as multiplying the expression by itself 2 times.

(3x^(2)*(x+10)(x+10))-x(x+10)^(3)=0

Multiply each term in the first group by each term in the second group using the FOIL method. FOIL stands for First Outer Inner Last, and is a method of multiplying two binomials. First, multiply the first two terms in each binomial group. Next, multiply the outer terms in each group, followed by the inner terms. Finally, multiply the last two terms in each group.

(3x^(2)(x*x+x*10+10*x+10*10))-x(x+10)^(3)=0

Simplify the FOIL expression by multiplying and combining all like terms.

(3x^(2)(x^(2)+20x+100))-x(x+10)^(3)=0

Multiply 3x^(2) by each term inside the parentheses.

((3x^(4)+60x^(3)+300x^(2)))-x(x+10)^(3)=0

Remove the parentheses around the expression 3x^(4)+60x^(3)+300x^(2).

(3x^(4)+60x^(3)+300x^(2))-x(x+10)^(3)=0

Cubing an expression is the same as multiplying the expression by itself 3 times.

(3x^(4)+60x^(3)+300x^(2))+(-x*(x+10)(x+10)(x+10))=0

Multiply each term in the first group by each term in the second group using the FOIL method. FOIL stands for First Outer Inner Last, and is a method of multiplying two binomials. First, multiply the first two terms in each binomial group. Next, multiply the outer terms in each group, followed by the inner terms. Finally, multiply the last two terms in each group.

(3x^(4)+60x^(3)+300x^(2))+(-x(x*x+x*10+10*x+10*10)(x+10))=0

Simplify the FOIL expression by multiplying and combining all like terms.

(3x^(4)+60x^(3)+300x^(2))+(-x(x^(2)+20x+100)(x+10))=0

Multiply each term in the first polynomial by each term in the second polynomial.

(3x^(4)+60x^(3)+300x^(2))+(-x(x^(2)*x+x^(2)*10+20x*x+20x*10+100*x+100*10))=0

Multiply each term in the first polynomial by each term in the second polynomial.

(3x^(4)+60x^(3)+300x^(2))+(-x(x^(3)+30x^(2)+300x+1000))=0

Multiply -x by each term inside the parentheses.

(3x^(4)+60x^(3)+300x^(2))+((-x^(4)-30x^(3)-300x^(2)-1000x))=0

Remove the parentheses around the expression -x^(4)-30x^(3)-300x^(2)-1000x.

(3x^(4)+60x^(3)+300x^(2))+(-x^(4)-30x^(3)-300x^(2)-1000x)=0

Remove the parentheses that are not needed from the expression.

3x^(4)+60x^(3)+300x^(2)-x^(4)-30x^(3)-300x^(2)-1000x=0

Since 3x^(4) and -x^(4) are like terms, add -x^(4) to 3x^(4) to get 2x^(4).

2x^(4)+60x^(3)+300x^(2)-30x^(3)-300x^(2)-1000x=0

Since 60x^(3) and -30x^(3) are like terms, add -30x^(3) to 60x^(3) to get 30x^(3).

2x^(4)+30x^(3)+300x^(2)-300x^(2)-1000x=0

Since 300x^(2) and -300x^(2) are like terms, add -300x^(2) to 300x^(2) to get 0.

2x^(4)+30x^(3)-1000x=0

Factor out the GCF of 2x from each term in the polynomial.

2x(x^(3))+2x(15x^(2))+2x(-500)=0

Factor out the GCF of 2x from 2x^(4)+30x^(3)-1000x.

2x(x^(3)+15x^(2)-500)=0

Factor the polynomial using the rational roots theorem.

2x(x-5)(x+10)(x+10)=0

Combine the two common factors of (x+10) by adding the exponents.

2x(x-5)(x+10)^(2)=0

Reorder the polynomial x-5 alphabetically from left to right, starting with the highest order term.

(2x*(x-5)*(x+10)^(2))=0

Multiply all the factors separated by a * in 2x*(x-5)*(x+10)^(2).

(2x(x-5)(x+10)^(2))=0

Squaring an expression is the same as multiplying the expression by itself 2 times.

((2x*(x-5)*(x+10)(x+10)))=0

Multiply all the factors separated by a * in 2x*(x-5)*(x+10)(x+10).

((2x(x-5)(x+10)(x+10)))=0

Multiply each term in the first group by each term in the second group using the FOIL method. FOIL stands for First Outer Inner Last, and is a method of multiplying two binomials. First, multiply the first two terms in each binomial group. Next, multiply the outer terms in each group, followed by the inner terms. Finally, multiply the last two terms in each group.

((2x(x*x+x*10-5*x-5*10)(x+10)))=0

Simplify the FOIL expression by multiplying and combining all like terms.

((2x(x^(2)+5x-50)(x+10)))=0

Multiply each term in the first polynomial by each term in the second polynomial.

((2x(x^(2)*x+x^(2)*10+5x*x+5x*10-50*x-50*10)))=0

Multiply each term in the first polynomial by each term in the second polynomial.

((2x(x^(3)+15x^(2)-500)))=0

Multiply 2x by each term inside the parentheses.

(((2x^(4)+30x^(3)-1000x)))=0

Remove the parentheses around the expression 2x^(4)+30x^(3)-1000x.

((2x^(4)+30x^(3)-1000x))=0

Remove the parentheses around the expression 2x^(4)+30x^(3)-1000x.

(2x^(4)+30x^(3)-1000x)=0

Factor out the GCF of 2x from each term in the polynomial.

(2x(x^(3))+2x(15x^(2))+2x(-500))=0

Factor out the GCF of 2x from 2x^(4)+30x^(3)-1000x.

(2x(x^(3)+15x^(2)-500))=0

Factor the polynomial using the rational roots theorem.

2x(x-5)(x+10)(x+10)=0

Combine the two common factors of (x+10) by adding the exponents.

2x(x-5)(x+10)^(2)=0

Set the single term factor on the left-hand side of the equation equal to 0.

2x=0

Divide each term in the equation by 2.

(2x)/(2)=(0)/(2)

Simplify the left-hand side of the equation by canceling the common terms.

x=(0)/(2)

0 divided by any number or variable is 0.

x=0

Set each of the factors of the left-hand side of the equation equal to 0.

x-5=0_x+10=0

Since -5 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 5 to both sides.

x=5_x+10=0

Set each of the factors of the left-hand side of the equation equal to 0.

x=5_x+10=0

Since 10 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 10 from both sides.

x=5_x=-10

The complete solution is the set of the individual solutions.

x=0,5,-10

Verify each of the solutions by substituting them back into the original equation ([3x(x+10)^(2)])/((x^(2)))=([(x+10)^(3)])/((x^(2))) and solving. In this case, the solution (0) was proven to be invalid during this process.

x=5,-10