Ernesto solves the equation below by first squaring both sides of the equation. sqrt((1)/(2)w+8)=-2 What extraneous solution does Ernesto obtain? w=

Write Equation: Write down the original equation. The original equation is sqrt((1)/(2)w+8)=-2.Square Both Sides: Square both sides of the equation to eliminate the square root. (sqrt((1)/(2)w+8))^2=(-2)^2 This gives us (1)/(2)w+8=4.Solve for w: Solve for w. Subtract 8 from both sides of the equation to isolate the term with w. (1)/(2)w+8-8=4-8 This simplifies to (1)/(2)w=-4.Multiply by 2: Multiply both sides by 2 to solve for w. 2*((1)/(2)w)=2*(-4) This simplifies to w=-8.Check for Extraneous Solutions: Check for extraneous solutions by substituting w back into the original equation. Substitute w=-8 into sqrt((1)/(2)w+8). sqrt((1)/(2)(-8)+8)=sqrt(-4+8)=sqrt(4) This simplifies to 2, not -2. Since the original equation has a negative square root, which is not possible, w=-8 is an extraneous solution.

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Q. Ernesto solves the equation below by first squaring both sides of the equation.

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\sqrt{\frac{1}{2} w+8}=-2

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What extraneous solution does Ernesto obtain?

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w=

Write Equation: Write down the original equation. $\newline$ The original equation is $\sqrt{\left(\frac{1}{2}w+8\right)}=-2$ .
Square Both Sides: Square both sides of the equation to eliminate the square root. $\newline$ $(\sqrt{\frac{1}{2}w+8})^2=(-2)^2$ $\newline$ This gives us $\frac{1}{2}w+8=4$ .
Solve for w: Solve for w. $\newline$ Subtract $8$ from both sides of the equation to isolate the term with $w$ . $\newline$ $\frac{1}{2}w+8-8=4-8$ $\newline$ This simplifies to $\frac{1}{2}w=-4$ .
Multiply by $2$ : Multiply both sides by $2$ to solve for $w$ . $2\left(\frac{1}{2}w\right)=2(-4)$ This simplifies to $w=-8$ .
Check for Extraneous Solutions: Check for extraneous solutions by substituting $w$ back into the original equation. $\newline$ Substitute $w=-8$ into $\sqrt{\left(\frac{1}{2}w+8\right)}$ . $\newline$ $\sqrt{\left(\frac{1}{2}(-8)+8\right)}=\sqrt{-4+8}=\sqrt{4}$ $\newline$ This simplifies to $2$ , not $-2$ . $\newline$ Since the original equation has a negative square root, which is not possible, $w=-8$ is an extraneous solution.