Question

If the sum of the roots of the equation \[{x^2} + px + q = 0\] is equal to the sum of their squares, then
A. \[{p^2} - {q^2} = Q\]
B. \[{p^2} + {q^2} = 2q\]
C. \[{p^2} + p = 2q\]
D. \[{q^2} + q = 2p\]

Answer 1

Hint: In this question, we have given that the sum of the roots of the given equation is equal to the sum of their squares. First of all, we will determine the sum and product of the roots of the given equation. After that, we'll apply the formula of the \[{\left( {\alpha + \beta } \right)^2}\] so we are going to place the worth of the sum and products of the roots of the equations. Hence, we will get our desired answer.

Formula Used:
Algebraic identity for square of the sum of two numbers:
\[{\left( {\alpha + \beta } \right)^2} = {\alpha ^2} + {\beta ^2} + 2\alpha \beta \]

Complete step by step solution:
We have been provided in the question that,
The sum of the roots of the equation \[{x^2} + px + q = 0\] is equal to the sum of their squares
Let us assume that the roots of the given equation are \[\alpha \] and \[\beta \]
\[{x^2} + px + q = 0\]
Now we have to determine the sum and the product of the roots of the given equation.
The sum of the roots can be written as
\[ \Rightarrow \alpha + \beta = - p\]
And the product of the root can be written as
\[ \Rightarrow \alpha \beta = q\]
Now according to the condition that is given in the question, the equation can be restructured as below
\[ \Rightarrow \alpha + \beta = {\alpha ^2} + {\beta ^2}\] -------- (1)
Now let us apply the formula of the \[{\left( {\alpha + \beta } \right)^2}\]
Therefore, the equation can be written as below
\[{\left( {\alpha + \beta } \right)^2} = {\alpha ^2} + {\beta ^2} + 2\alpha \beta \]
Now we are going to place the value of the equation (1) in the above equation
Therefore, we are going to get
\[{\left( {\alpha + \beta } \right)^2} = \left( {\alpha + \beta } \right) + 2\alpha \beta \]
Now we have obtained both the values of the \[\left( {\alpha + \beta } \right)\] and \[\alpha \beta \]
Therefore, we will obtain
\[ \Rightarrow {( - p)^2} = ( - p) + 2q\]
On rewriting the above equation in quadratic form we get,
\[ \Rightarrow {p^2} + p - 2q = 0\]
Therefore, if the sum of the roots of the equation \[{x^2} + px + q = 0\] is equal to the sum of their squares, then \[{p^2} + p - 2q = 0\]
Hence, the option C is correct

Note: When using the relationship between the coefficients and roots of the polynomial, be mindful of the signs. You should also understand all of the algebraic and exponential formulas because they are frequently employed.