How many solutions does 1/[x*(x + 1)] + 1/[x*(x + 2)] = 1/(x +1) have?

Question

How many solutions does  \frac{1}{x*(x+1)}+\frac{1}{x*(x+2)}=\frac{1}{x+1}  have?

A. None
B. 1
C. 2
D. 3
E. 5

PyjamaScientist · Answer

Option (C)   : Two roots (- \sqrt{3} ,  \sqrt{3} )

Upon simplifying, you will get:

x^2 - 3= 0

This gives two solutions, +/-  \sqrt{3} .

JerryAtDreamScore · Answer

Breaking Down the Info:

Multiply both sides by  x*(x+1)*(x + 2) . This gives us:

x + 2 + x + 1 = x*(x + 2)

x^2 - 3 = 0

That clearly gives two solutions (and we don't need to check denominators since the solutions are irrational).

Answer: C

Fdambro294 · Answer

Moving through the Algebra:

Move the leftward term to the right hand side

1 / (x) (x + 2) = 1 / (x + 1) - 1 / (x) (x + 1)

Subtract the two terms on the right hand side ——-> LCD = (x) (x + 1)

1 / (x) (x + 2) = X / (x) (x + 1) - 1 / (x) (x + 1)

1 / (x) (x + 2) = (x - 1) / (x) (x + 1)

Cross multiply

(x) (x + 1) = (x) (x + 2) (x - 1)

(x) (x + 1) - (x) (x + 2) (x - 1) = 0

Take common factor (x)

(X) (x + 1 - (x + 2) (x - 1) ) = 0

(X) (x + 1 - x^2 - x + 2) = 0

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

X can not equal 0 because it will result in an undefined fraction (0 can not be in the denominator)

Therefore

(3 - x^2) must = 0

x^2 = 3

There are 2 roots:

+positive square root of 3

-negative square root of 3

2

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luisdicampo · Answer

Deconstructing the Question

We need to find how many solutions satisfy

\frac{1}{x(x+1)}+\frac{1}{x(x+2)}=\frac{1}{x+1}

First check the restricted values. Since they make a denominator zero,  x  cannot be

0,\ -1,\ -2

Then clear the denominators and solve the resulting equation.

Step-by-step

Start with

\frac{1}{x(x+1)}+\frac{1}{x(x+2)}=\frac{1}{x+1}

Multiply every term by the common denominator

x(x+1)(x+2)

The first term becomes

x+2

The second term becomes

x+1

The right side becomes

x(x+2)

So the equation becomes

(x+2)+(x+1)=x(x+2)

Simplify:

2x+3=x^2+2x

Subtract  2x  from both sides:

3=x^2

So

x^2=3

This gives

x=\sqrt{3}

or

x=-\sqrt{3}

Neither value is forbidden, so both are valid solutions.

Answer: C

How many solutions does 1/[x(x + 1)] + 1/[x(x + 2)] = 1/(x +1) have?

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