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 + 7} = x + 5\\) is to square both sides, obtaining \\(x + 7 = x^2 + 25\\). 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 + 7} = x + 5\\) is to square both sides, obtaining \\(x + 7 = \square\\). (simplify your answer. do not factor.) \\(\bigcirc\\) b. the statement is false because the first step in solving \\(\sqrt{x + 7} = x + 5\\) is to square both sides, obtaining \\((x + 7)^2 = \square\\). (simplify your answer. do not factor.) \\(\bigcirc\\) c. the statement is false because the first step in solving \\(\sqrt{x + 7} = x + 5\\) is to get the x - terms on one side and the constants on the other. \\(\bigcirc\\) d. the statement is true.

Explanation:

Step1: Recall squaring both sides

To solve an equation with a square root like \(\sqrt{x + 7}=x + 5\), we square both sides. The left side is \(\sqrt{x + 7}\), so squaring it gives \(x + 7\). The right side is \(x + 5\), so squaring it gives \((x + 5)^2\). Let's expand \((x + 5)^2\) using the formula \((a + b)^2=a^2+2ab + b^2\), where \(a = x\) and \(b = 5\). So \((x + 5)^2=x^2+10x + 25\).

Step2: Analyze the given statement

The original statement says squaring both sides gives \(x + 7=x^2+25\), but we know that \((x + 5)^2=x^2+10x + 25\), not \(x^2+25\). So the first step of squaring both sides should give \(x + 7=(x + 5)^2=x^2+10x + 25\). So option A is the correct type of error, and we need to find what \(x + 7\) equals, which is \((x + 5)^2=x^2+10x + 25\).

Answer:

A. The statement is false because the first step in solving \(\sqrt{x + 7}=x + 5\) is to square both sides, obtaining \(x + 7=\boldsymbol{x^{2}+10x + 25}\).