Question

The square root of \[{x^4} + 6{x^3} + 17{x^2} + 24x + 16\] is equal to \[{x^2} + 3x + 4\]
A.True
B.False

Answer 1

Hint: To solve this question we will take any general quadratic equation then square the equation. After rewriting the terms of that equation and comparing it with the given equation in question. If we get the same quadratic equation mentioned in the question, then we can state that the given statement is true otherwise it is false.

Complete step by step solution:
Let the square root of the given equation be \[{x^2} + px + q\].
Now we will square the above equation. Therefore, we get
\[ \Rightarrow {\left( {{x^2} + px + q} \right)^2} = {x^4} + {p^2}{x^2} + {q^2} + 2p{x^3} + 2pqx + 2q{x^2}\]
Simplifying the above expression, we get
\[ \Rightarrow {\left( {{x^2} + px + q} \right)^2} = {x^4} + {x^3}\left( {2p} \right) + {x^2}\left( {{p^2} + 2q} \right) + x\left( {2pq} \right) + {q^2}\]
The given equation in the question is \[{x^4} + 6{x^3} + 17{x^2} + 24x + 16\].
Now we will compare this obtained equation with the given equation to find the square root of the given equation. Thus by comparing these two equations, we get
\[\begin{array}{l}2p = 6\\ \Rightarrow p = 3\end{array}\]
Also, we get
\[\begin{array}{l}{q^2} = 16\\ \Rightarrow q = 4\end{array}\]
Now we will put these values in the square root equation to get the required square root of the given equation. Therefore, we get
\[{x^2} + px + q = {x^2} + 3x + 4\]
Hence, the square root of the given equation is \[{x^2} + 3x + 4\] which is the same as mentioned in the problem.
Therefore, we can say that the given statement is true.
Hence, option (A) is the correct option.

Note: Quadratic equation is an equation, which has the highest degree of the variable is 2. A quadratic equation has only two roots and not more than that. Here, we could have directly found out the square of \[{x^2} + 3x + 4\] and if the square would have been equal to \[{x^4} + 6{x^3} + 17{x^2} + 24x + 16\], then we can easily say that \[{x^2} + 3x + 4\] is the square root of the given expression.