Question

Find ‘m’ for which \[m{{x}^{2}}+\left( 2m-1 \right)x+m-1=0\] has roots of opposite sign?

Answer 1

Hint: In order to solve the given question for the value of ‘m’, first we need to simplify the given equation so that we will get two roots of the equation i.e. the value of ‘x’. Later after getting the one value of ‘x’, it is given in the question that the other root is of the opposite sign. By changing the sign of the root and putting it in the equation will get the required value of ‘m’.

Complete step-by-step answer:
We have given an equation,
 \[m{{x}^{2}}+\left( 2m-1 \right)x+m-1=0\]
Simplifying the above equation, we will get
 \[m{{x}^{2}}+2mx-x+m-1=0\]
It can be rewrite as,
 \[m{{x}^{2}}+mx+mx-x+m-1=0\]
Taking out the common terms and writing it in pairs,
We will obtain
 \[mx\left( x+1 \right)+x\left( m-1 \right)+1\left( m-1 \right)=0\]
It can be rewrite as,
 \[mx\left( x+1 \right)+\left( x+1 \right)\left( m-1 \right)=0\]
Taking out \[\left( x+1 \right)\] as a common factor, we will get
 \[\left( x+1 \right)\left( mx+\left( x+1 \right) \right)=0\]
Hence,
Equation the factors \[\left( x+1 \right)\] equal to zero, we will obtain
 \[\Rightarrow \left( x+1 \right)=0\ \]
Solving the above for the value of ‘x’, we will get
 \[\Rightarrow x=-1\ \ \]
Since we get the value of the one root i.e. x = -1.
It is given in the question that the roots of the given equation is of opposite sign.
Hence,
The other root will be;
 \[\Rightarrow x=1\ \ \]
Now,
We have the equation,
 \[\Rightarrow \left( x+1 \right)\left( mx+\left( x+1 \right) \right)=0\]
Putting \[x=1\ \ \] in the above equation,
 \[\Rightarrow \left( 1+1 \right)\left( m\left( 1 \right)+1+1 \right)=0\ \]
Solving the above, we will get
 \[\Rightarrow 2\left( m+2 \right)=0\ \]
Solving the above using the distributive property i.e. \[a\left( b+c \right)=ab+ac\] .
 \[\Rightarrow 2m+4=0\ \]
Subtracting 4 from both the sides of the equation, we will get
 \[\Rightarrow 2m=-4\]
Dividing both the sides of the equation by 2, we will get
 \[\Rightarrow m=\dfrac{-4}{2}=-2\]
Therefore,
 \[\Rightarrow m = -2\]
So, the correct answer is “ m = -2”.

Note: While solving these types of questions it is important to convert the equation into its quadratic form so that we can easily simplify the given equation and find the values of both the roots. Students need to do all the simplification and the calculation part very carefully and explicitly to avoid making any type of error as it will give you the incorrect answer i.e. wrong value of ‘m’.