Question

If \[3 + 4i\] is a root of the equation \[{x^2} + px + q = 0\] (\[p,{\text{ }}q\] are real numbers), then
A) \[p = 6,q = 25\]
B) \[p = 6,q = 1\]
C) \[p = - 6,q = - 7\]
D) \[p = - 6,q = 25\]

Answer 1

Hint: In this question, we are given that \[3 + 4i\] is the root of the equation \[{x^2} + px + q = 0\] and we have to calculate the value of \[p,{\text{ }}q\]. Firstly, the other root of the equation will be the conjugate of the first root. Then, calculate the sum and product of the roots using the formula $\alpha + \beta = \dfrac{{ - B}}{A}$, $\alpha \beta = \dfrac{C}{A}$ where the equation is \[A{x^2} + Bx + C = 0\].

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, $\alpha \beta = \dfrac{C}{A}$
Conjugate complex numbers –
$\overline {a + ib} = a - ib$

Complete step by step Solution:
Given that,
\[3 + 4i\] is the root of the equation \[{x^2} + px + q = 0\].
As we know, when a complex integer is a root of a polynomial with real coefficients, its complex conjugate is likewise a root. A complex conjugate is a number that has an equally real part and an imaginary part that is equal in magnitude but opposite in sign.
It implies that the other root of the equation is the conjugate of \[3 + 4i\] i.e., \[3 - 4i\]
Compare the given equation \[{x^2} + px + q = 0\] with the general quadratic equation \[A{x^2} + Bx + C = 0\]
We get, $A = 1,B = p,C = q$
Now, applying the formula of sum and product of the roots
Therefore,
Sum of the roots $ = \dfrac{{ - B}}{A}$
$ \Rightarrow 3 + 4i + 3 - 4i = - p$
On simplifying, we get $p = - 6$
Now, the product of roots $ = \dfrac{C}{A}$
$ \Rightarrow \left( {3 + 4i} \right)\left( {3 - 4i} \right) = q$
Using the algebraic identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
It implies that,
$q = 9 - 16{i^2}$
Also, the square of $i$ is equal to $ - 1$
So, $q = 25$
Thus, the value of $p,q$ are $ - 6,25$ respectively.

Hence, the correct option is (D).

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. Also, in this question, we must compute the product and sum of the new roots and express them in terms of the coefficients of the provided equation, $a,b$, and $c$. ${x^2} + Sx + P = 0$ is the quadratic equation with sum of roots $S$ and product of roots $P$. We must remember to add an extra negative sign to the negative sign in the sum of the roots when substituting the sum and product. We must ensure that the denominators of the sum and the product of the roots are the same so that we can cancel the denominator by multiplying them with the denominator throughout the equation.