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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=

Ernesto solves the equation below by first squaring both sides of the equation.\newline12w+8=−2 \sqrt{\frac{1}{2} w+8}=-2 \newlineWhat extraneous solution does Ernesto obtain?\newlinew= w=

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Q. Ernesto solves the equation below by first squaring both sides of the equation.\newline12w+8=−2 \sqrt{\frac{1}{2} w+8}=-2 \newlineWhat extraneous solution does Ernesto obtain?\newlinew= w=
  1. Write Equation: Write down the original equation.\newlineThe original equation is (12w+8)=−2\sqrt{\left(\frac{1}{2}w+8\right)}=-2.
  2. Square Both Sides: Square both sides of the equation to eliminate the square root.\newline(12w+8)2=(−2)2(\sqrt{\frac{1}{2}w+8})^2=(-2)^2\newlineThis gives us 12w+8=4\frac{1}{2}w+8=4.
  3. Solve for w: Solve for w.\newlineSubtract 88 from both sides of the equation to isolate the term with ww.\newline12w+8−8=4−8\frac{1}{2}w+8-8=4-8\newlineThis simplifies to 12w=−4\frac{1}{2}w=-4.
  4. Multiply by 22: Multiply both sides by 22 to solve for ww.2(12w)=2(−4)2\left(\frac{1}{2}w\right)=2(-4)This simplifies to w=−8w=-8.
  5. Check for Extraneous Solutions: Check for extraneous solutions by substituting ww back into the original equation.\newlineSubstitute w=−8w=-8 into (12w+8)\sqrt{\left(\frac{1}{2}w+8\right)}.\newline(12(−8)+8)=−4+8=4\sqrt{\left(\frac{1}{2}(-8)+8\right)}=\sqrt{-4+8}=\sqrt{4}\newlineThis simplifies to 22, not −2-2.\newlineSince the original equation has a negative square root, which is not possible, w=−8w=-8 is an extraneous solution.

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