Question

Solve the equation for: \[(p - q){x^2} + 5(p - q)x - 2(p - q) = r\] given that, p, q and r are real.
(a) $p + q$
(b) $p - q$
(c) Cannot be determined
(d) None of these

Answer 1

Hint: We will assume the given parameters p, q and r as any constant comparing it with the standard quadratic equation that is \[{x^2} + 2xy + {y^2} = 0\] we can find the required roots. But, due to unknown factors defined in the problem the roots cannot be determined, respectively. Also, the problem can be solved by another method that is simplifying the equation mathematically as per BODMAS, and at a certain stage the solution will be in the “cannot determined stage”!

Complete answer:The given equation is,
\[(p - q){x^2} + 5(p - q)x - 2(p - q) = r\]
Where, p, q, r are real
$\therefore $Mathematically, the equation becomes,
\[(p - q){x^2} + 5(p - q)x - 2(p - q) - r = 0\]
\[(p - q){x^2} + 5(p - q)x - \left[ {2(p - q) + r} \right] = 0\]
Since, substituting$p - q = A$, we get
Where, ‘A’ is pure real number, … ($\because $given)
The equation becomes
\[A{x^2} + 5Ax - \left( {2A + r} \right) = 0\]
Using the factorization method that is $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, we get
$x = \dfrac{{ - 5A \pm \sqrt {25{A^2} - 4A\left[ { - \left( {2A + r} \right)} \right]} }}{{2A}}$
Solving the equation mathematically, we get
$x = \dfrac{{ - 5A \pm \sqrt {25{A^2} + 8{A^2} + 4Ar} }}{{2A}}$
Adding the certain terms, we get
$x = \dfrac{{ - 5A \pm \sqrt {33{A^2} + 4Ar} }}{{2A}}$
Re-substituting the value of ‘A’, we get
$x = \dfrac{{ - 5\left( {p - q} \right) \pm \sqrt {33{{\left( {p - q} \right)}^2} + 4\left( {p - q} \right)r} }}{{2\left( {p - q} \right)}}$
But, since given equation is uncertain with unknown values or general variables used,
So, it is not possible to determine the exact roots of the given equation as the values are general in terms of algebraic variables that is ‘$p,q,r$’ respectively.
As a result, we can only determine that the roots will real that is given to us as the condition,
Therefore, it seems that predominantly we cannot find the exact roots of the equation!
$\therefore \Rightarrow $Hence, option (c) is correct.

Note:
 One must remember the basic algebraic formulae or identities such as ${(x + y)^2} = {x^2} + 2xy + {y^2}$, ${(x - y)^2} = {x^2} - 2xy + {y^2}$, etc. so as to solve the problem efficiently. Here, in this case, the values are unknown, so it is impossible to solve the required equation. Algebraic identities play a significant role in solving this problem.