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 + 8} = x + 4\\) is to square both sides, obtaining \\(x + 8 = x^2 + 16\\). 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 + 8} = x + 4\\) is to get the \\(x\\)-terms on one side and the constants on the other. \\(\bigcirc\\) b. the statement is false because the first step in solving \\(\sqrt{x + 8} = x + 4\\) is to square both sides, obtaining \\(x + 8 = \square\\) (simplify your answer. do not factor.) \\(\bigcirc\\) c. the statement is false because the first step in solving \\(\sqrt{x + 8} = x + 4\\) is to square both sides, obtaining \\((x + 8)^2 = \square\\). (simplify your answer. do not factor.) \\(\bigcirc\\) d. the statement is true.

Explanation:

Step1: Recall squaring both sides

To solve an equation with a square root like \(\sqrt{x + 8}=x + 4\), the first step is to square both sides. The right - hand side is \((x + 4)\), and when we square \((x + 4)\) using the formula \((a + b)^2=a^{2}+2ab + b^{2}\) where \(a=x\) and \(b = 4\), we get \((x + 4)^{2}=x^{2}+8x + 16\).

The original statement claims that squaring both sides gives \(x + 8=x^{2}+16\), which is incorrect. The correct result of squaring both sides is \(x + 8=(x + 4)^{2}=x^{2}+8x + 16\).

Step2: Analyze the options

• Option A: The first step for an equation with a square root is not to get \(x\) - terms and constants on different sides. The first step is to isolate the square root (which is already isolated here) and then square both sides. So A is incorrect.
• Option B: When we square both sides of \(\sqrt{x+8}=x + 4\), we have \(x + 8=(x + 4)^{2}=x^{2}+8x + 16\). So the correct expression after squaring both sides is \(x^{2}+8x + 16\).
• Option C: We square both sides of \(\sqrt{x + 8}=x + 4\), so we square the left - hand side \(\sqrt{x + 8}\) to get \(x + 8\) and the right - hand side \(x + 4\) to get \((x + 4)^{2}\), not \((x + 8)^{2}\). So C is incorrect.
• Option D: The statement is false because when we square both sides of \(\sqrt{x+8}=x + 4\), we get \(x + 8=(x + 4)^{2}=x^{2}+8x + 16\), not \(x + 8=x^{2}+16\).

Answer:

B. The statement is false because the first step in solving \(\boldsymbol{\sqrt{x + 8}=x + 4}\) is to square both sides, obtaining \(\boldsymbol{x + 8=x^{2}+8x + 16}\)