Question

If the roots of the equation \[(a - b){x^2} + (b - c)x + (c - a) = 0\] are equal, prove that \[2a = b + c\]

Answer 1

Hint: The question is related to the quadratic equation and here we have to prove the given condition. Since they have mentioned the nature of the root of the quadratic equation, we use the formula of a discriminant and hence we can prove the given condition.

Complete step-by-step answer:
The equation is of the form of a quadratic equation. In general the quadratic equation will be in the form of \[a{x^2} + bx + c\]. The discriminant is the part of the quadratic formula underneath the square root symbol: \[{b^2} - 4ac\]. The discriminant tells us whether there are two solutions, one solution, or no solutions.
If the roots are equal the discriminant is equal to zero. If the roots are positive real the discriminant is greater than zero. If the roots are negative the discriminant is less than zero.
Now on considering the given question
\[ \Rightarrow (a - b){x^2} + (b - c)x + (c - a) = 0\]
The above equation is a quadratic equation. Here the value of \[a = (a - b)\], \[b = (b - c)\] and \[c = (c - a)\] on comparing the given equation to a general form of equation.
Here the roots are equal then the discriminant is equal to zero
So we have
\[ \Rightarrow D = 0\]
The discriminant is given by \[D = {b^2} - 4ac\]
On substituting the known values in the discriminant formula
\[ \Rightarrow D = {(b - c)^2} - 4(a - b)(c - a)\]
On simplifying
\[ \Rightarrow {b^2} + {c^2} - 2bc - 4(ac - {a^2} - bc + ab)\]
\[ \Rightarrow {b^2} + {c^2} - 2bc - 4ac + 4{a^2} + 4bc - 4ab\]
\[ \Rightarrow 4{a^2} + {b^2} + {c^2} + 2bc - 4ac - 4ab\]
\[ \Rightarrow {\left( {2a - b - c} \right)^2}\]
Here discriminant is zero so we equate the above term to zero so we have
\[ \Rightarrow {\left( {2a - b - c} \right)^2} = 0\]
Taking square root on both sides we get
\[ \Rightarrow 2a - b - c = 0\]
\[ \Rightarrow 2a = b + c\]
Hence proved

Note: The roots depend on the value of the discriminant. The discriminant is given by \[{b^2} - 4ac\]. The discriminant tells us whether there are two solutions, one solution, or no solutions. While simplifying we use the simple arithmetic operations. hence we obtain the required solution.