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.

Return to Assignments Equations with fractions

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Solution assignment 07 Equations with fractions

Assignment 7

Solution

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