Question

Error analysis: You are going to find the error(s) in the math work shown below. Explain any errors completely, then enter the correct solution including the correct steps. For both problems thank you

Answer 1

Answer:

  2. (√30)/2

  3. x = ±2

Step-by-step explanation:

For these, we will show the correct steps, and describe the error in words.

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2. Rationalize the denominator

  $\displaystyle\begin{aligned}\sqrt{\dfrac{15}{2}&=\dfrac{\sqrt{15}}{\sqrt{2}}\\&=\dfrac{\sqrt{15}}{\sqrt{2}}}\cdot\dfrac{\sqrt{2}}{\sqrt{2}}}\\ &=\dfrac{\sqrt{30}}{2}&\text{error in original work was here}\end{aligned}$

The product of √2 and itself is 2, not √2.

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3. Solve by taking square root

  $\displaystyle\begin{aligned}x^2-4&=0\\x^2&=4&\text{error was here}\\x&=\pm\sqrt{4}&\text{both square roots}\\x&=\pm2\end{aligned}$

The expression (x²-4) does not have a square root of (x-2). To usefully take the square root, we need a perfect square involving the variable term.

Answer 2

2) Rationalize the denominator:

$\sf \sqrt{\dfrac{15}{2} } = \dfrac{\sqrt{15} }{\sqrt{2} }$

        $\sf = \dfrac{\sqrt{15} }{\sqrt{2} } \ \cdot \ \dfrac{\sqrt{2} }{\sqrt{2} }$        

        $\sf = \dfrac{\sqrt{15 * 2} }{\sqrt{2} * \sqrt{2} }$                [ apply radical rule: √a × √b = √ab ]

        $\sf = \dfrac{\sqrt{30} }{2}$                      [multiply the integers] [note: √2 × √2 is 2]

3) Solve by taking the square root:

$\sf x^2 -4 = 0 \\\\\sqrt{x^2-4} = 0\\ \\x^2 - 2^2 = 0^2 \ \quad \quad \quad \leftarrow third \ step \ should \ be \ this\\\\ (x-2)(x + 2) = 0 \\\\x = 2, x = -2 \\ \\ x = \pm 2$