Question

If $p$ and $q$ are the roots of the equation \[{x^2} + pq = (p + 1)x\], then $q = $
A. $ - 1$
B. $1$
C. $2$
D. None of these

Answer 1

Hint: In this question, we are given a quadratic equation \[{x^2} + pq = (p + 1)x\] whose roots are $p$ and $q$. We have to calculate the value of $q$. Compare the equation with the general quadratic equation \[A{x^2} + Bx + C = 0\] and then apply the formula of sum and product of the roots i.e., $\alpha + \beta = \dfrac{{ - B}}{A}$ and $\alpha \beta = \dfrac{C}{A}$.

Formula Used:
General quadratic equation: – \[A{x^2} + Bx + C = 0\]
Let, the roots of the above quadratic equation be $\alpha $ and $\beta $
Therefore,
Sum of roots, $\alpha + \beta = \dfrac{{ - B}}{A}$
Product of roots, $\alpha \beta = \dfrac{C}{A}$

Complete step by step Solution:
Given that,
\[{x^2} + pq = (p + 1)x\], a quadratic equation
Also written as, \[{x^2} - (p + 1)x + pq = 0\]
Compare the above equation \[{x^2} - (p + 1)x + pq = 0\] with the general quadratic equation \[A{x^2} + Bx + C = 0\]
We get, $A = 1$, $B = - \left( {p + 1} \right)$, and $C = pq$
Now, we are given that $p$ and $q$ are the roots of \[{x^2} - (p + 1)x + pq = 0\]
Using the formula of sum and product of the roots
Therefore,
$p + q = p + 1$ and $pq = pq$
On comparing both the sides of the equation of the sum of roots $p + q = p + 1$
It implies that, $q = 1$
Thus, the value of $q$ is $1$.

Hence, the correct option is (B).

Additional Information:In algebra, a quadratic equation is any equation that may be rewritten in a standard form or the equation of the ${2^{nd}}$ degree. That is, it will contain at least one squared phrase. In the equation a \[A{x^2} + Bx + C = 0\], for example, $x$ represents an unknown integer, whereas $a,b,$ and $c$ represent known values or numerical coefficients, where $a = 0$.


Note: The values of $x$ that fulfil a given quadratic equation \[A{x^2} + Bx + C = 0\] are known as its roots. They are, in other words, the values of the variable $\left( x \right)$ that satisfy the equation. The roots of a quadratic function are the $x - $ coordinates of the function's $x - $ intercepts. Because the degree of a quadratic equation is $2$, it can only have two roots.