IMAT 2018 Q53 [Sum of Roots]

The equation below has two roots.

x= \frac{x+4}{x+1}

What is the sum of the two roots?

A. 2
B. – 2
C. 4
D. – 4
E. 0

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In order to solve this, firstly we need to find the equation in the form f(x)=0 and calculate its roots.

x = \frac{x+4}{x+1} \to x \cdot (x+1) = x+4 \to x^2 + x -(x+4) =0 \to
\to x^2 + x -x-4 =0 \to x^2 -4 =0 \to x^2 = 4 \to x = \frac{+}{} 2

From this, we can see that the sum of the roots is 0, so the correct answer is (E)

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We can also solve this question by using Vieta’s formulas:

From the quadric equation in the previous comment: x^2-4=0. Comparing it with the definition: ax^2+bx+c=0, we can see that a=1, b=0 and c=-4.

Vieta’s formulas say: x_1+x_2 =\frac{-b}{a}. In our case, b=0, so the value of the sum is equal to 0.