In Exercises 129-132, determine whether each statement is...

In Exercises $129-132$, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
To solve $x-9 \sqrt{x}+14=0,$ we let $\sqrt{u}=x$

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Key Concepts

Substitution Method in Solving Equations

This technique involves replacing a complicated expression, such as a radical, with a new variable to transform the equation into a simpler form, often a polynomial. Once the simpler equation is solved, the substitution is reversed to determine the original variable's values.

Variable Transformation

Variable transformation is the act of defining a new variable in terms of an existing expression to simplify the structure of an equation. It is crucial that the new variable is defined correctly to maintain the equivalence between the transformed equation and the original one.

Correct Identification of the Substituted Expression

When performing substitutions, especially in equations involving radicals, it is important to substitute the expression that actually appears in the equation. For instance, if an equation involves ?x, a common strategy is to set u = ?x rather than setting ?u = x, to properly reduce the equation into a standard quadratic form.

Domain Restrictions for Radical Functions

Equations involving square roots have inherent domain restrictions since the expression under a square root must be non-negative. When solving these equations, one must consider the valid domain of the substituted variable to avoid introducing extraneous solutions that do not satisfy the original equation.

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