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Which of the following statements are true about the given rational equation?

4/x+6 +1/x²=x+10/x³+6x²
Check all of the boxes that apply.
A. x = 1 is a solution.
B. x = 0 is an extraneous solution.=
C. x = –1 is a solution.
D. x = –6 is an extraneous solution.

User Joanbm
by
8.2k points

2 Answers

2 votes

Answer:

A and C is correct.

Explanation:


\text{Rational Equation: }(4)/(x+6)+(1)/(x^2)=(x+10)/(x^3+6x^2)

First we simplify this equation (-1 + x^2)/(x (6 + x))


(x^2-1)/(x(x+6))=0

Solution set: x=1,-1

We have to check all option which is solution of above equation.

A) x=1, we will put x=1 into equation both sides.


\text{Left Side:} \Rightarrow (4)/(x+6)+(1)/(x^2)


\Rightarrow (4)/(1+6)+(1)/(1^2)


\Rightarrow (11)/(7)


\text{Right Side:} \Rightarrow (x+10)/(x^3+6x^2)


\Rightarrow (1+10)/(1^3+6* 1^2)


\Rightarrow (11)/(7)


\text{Left Side}=\text{Right Side}=(11)/(7)

Thus, x=1 is a solution.

C) x=-1, we will put x=-1 into equation both sides.


\text{Left Side:} \Rightarrow (4)/(x+6)+(1)/(x^2)


\Rightarrow (4)/(-1+6)+(1)/((-1)^2)


\Rightarrow (9)/(5)


\text{Right Side:} \Rightarrow (-1+10)/(-1^3+6(-1)^2)


\Rightarrow (9)/(-1+6* 1)


\Rightarrow (9)/(5)


\text{Left Side}=\text{Right Side}= (9)/(5)

Thus, x=-1 is a solution.

Thus, A and C is correct.


User Wunch
by
8.1k points
2 votes

Answer:

Correct choices are A and C

Explanation:

Consider the equation


(4)/(x+6)+(1)/(x^2)=(x+10)/(x^3+6x^2).

Note that
x\\eq 0,\ x\\eq-6.

The left side of this equation is equal to


(4)/(x+6)+(1)/(x^2)=(4\cdot x^2+1\cdot (x+6))/((x+6)x^2)=(4x^2+x+6)/(x^3+6x^2).

Since the left and the right sides of this equation have the same denominators, their numerators are equal


4x^2+x+6=x+10.

Then


4x^2=10-6,\\ \\4x^2=4,\\ \\x^2=1,\\ \\x_1=1,\ x_2=-1.

Both these numbers are the solutions of the given equation.

User Scott Marchant
by
8.0k points
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