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