Question

determine whether the following statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement. the first step in solving \\(\sqrt{x + 5} = x + 2\\) is to square both sides, obtaining \\(x + 5 = x^2 + 4\\) select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. the statement is false because the first step in solving \\(\sqrt{x + 5} = x + 2\\) is to square both sides, obtaining \\(x + 5 = \square\\) (simplify your answer. do not factor.) \\(\bigcirc\\) b. the statement is false because the first step in solving \\(\sqrt{x + 5} = x + 2\\) is to square both sides, obtaining \\((x + 5)^2 = \square\\) (simplify your answer. do not factor.) \\(\bigcirc\\) c. the statement is false because the first step in solving \\(\sqrt{x + 5} = x + 2\\) is to get the \\(x\\)-terms on one side and the constants on the other \\(\bigcirc\\) d. the statement is true

Explanation:

Step1: Square the right-hand side

When squaring \(x+2\), use the formula \((a+b)^2 = a^2 + 2ab + b^2\).
$$(x+2)^2 = x^2 + 2(x)(2) + 2^2 = x^2 + 4x + 4$$

Step2: Compare with given result

The original statement claims squaring gives \(x+5=x^2+4\), which is wrong because the right-hand side was expanded incorrectly.

Answer:

A. The statement is false because the first step in solving $\sqrt{x+5} = x+2$ is to square both sides, obtaining $x+5 = x^2 + 4x + 4$