Solve Lesson 39 Page 60 SBT Math 10 – Kite>


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 \(\sqrt {f\left( x \right)} = g\left( x \right)\)



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