Solution assignment 07 Equations with fractions

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Assignment 7

For which value(s) of p does the following equation have two different solutions?

\displaystyle\frac{2x+p}{x-1}=\displaystyle\frac{x}{x+1}

Solution

Cross-multiplication yields:

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

2x^2+(p+2)x+p=x^2-x

x^2+(p+3)x+p=0

This is a quadratic equation in x. In order to find out whether this equation has two different solutions we look at the discriminant which has to be greater than 0:

D=(p+3)^2-4p>0

p^2+6p+9-4p>0

p^2+2p+9>0

The graph of the function in p in the left-hand side is an ‘opens up’ parabola lying above the horizontal axis because the discriminant is less than 0:

D=4-36<0

We may conclude that the inequality in p holds for all p and thus that the equation in x has two solutions for all values of p.

Note that we used a discriminant twice. In one case we needed the discriminant to determine whether the equation in x has two different solutions. In the other case we needed the discriminant to determine whether the graph of the function in p lies above the P-axis.

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