Topic
Explain why it is only necessary to check the solution of the equation \(f\left( x \right) = {\left[ {g\left( x \right)} \right]^2}\) satisfy the inequality \(g\left( x \right) \ge 0\) without checking the inequality \(f\left( x \right) \ge 0\) to conclude the solution of the equation \(\sqrt {f\left( x \right)} = g\left( x \right)\)
Solution method – See details
\(\sqrt {f\left( x \right)} = g\left( x \right) \Leftrightarrow \left\{ \begin{array}{l}g\left( x \right) \ge 0\\ f\left( x \right) = {\left[ {g\left( x \right)} \right]^2}\end{array} \right.\)
Detailed explanation
\(\sqrt {f\left( x \right)} \ge 0 \Rightarrow g\left( x \right) \ge 0\) Then \(f\left( x \right) = {\left[ {g\left( x \right)} \right]^2} \ge 0\), satisfying the CKD of the root.
We have \(\sqrt {f\left( x \right)} = g\left( x \right) \Leftrightarrow \left\{ \begin{array}{l}g\left( x \right) \ge 0 \\f\left( x \right) = {\left[ {g\left( x \right)} \right]^2}\end{array} \right.\)
So just check the solution of the equation \(f\left( x \right) = {\left[ {g\left( x \right)} \right]^2}\) satisfy the inequality \(g\left( x \right) \ge 0\) without checking the inequality \(f\left( x \right) \ge 0\) to conclude the solution of the equation \(\sqrt {f\left( x \right)} = g\left( x \right)\)
