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Chun was asked whether the following equation is an identity:

3(2x-1)^(2)+27=36x^(2)-36 x+36
He performed the following steps:

3(2x-1)^(2)+27

longrightarrow^(" Step "1)=3(2x-1)(2x-1)+27

^(" Step ")2=(6x-3)(6x-3)+27

^(" Step ")3=36x^(2)-18 x-18 x+9+27

^(" Step "4)=36x^(2)-36 x+36
For this reason, Chun stated that the equation is a true identity.
Is Chun correct? If not, in which step did he make a mistake?
Choose 1 answer:
(A) Chun is correct.
(B) Chun is incorrect. He made a mistake in step 2.
(C) Chun is incorrect. He made a mistake in step 3.
(D) Chun is incorrect. He made a mistake in step 4.

Chun was asked whether the following equation is an identity:\newline3(2x1)2+27=36x236x+36 3(2 x-1)^{2}+27=36 x^{2}-36 x+36 \newlineHe performed the following steps:\newline3(2x1)2+27 3(2 x-1)^{2}+27 \newline Step 1=3(2x1)(2x1)+27 \stackrel{\text { Step } 1}{\hookrightarrow}=3(2 x-1)(2 x-1)+27 \newline Step 2=(6x3)(6x3)+27 \stackrel{\text { Step } 2}{\hookrightarrow}=(6 x-3)(6 x-3)+27 \newline Step 3=36x218x18x+9+27 \stackrel{\text { Step } 3}{\hookrightarrow}=36 x^{2}-18 x-18 x+9+27 \newline Step 4=36x236x+36 \stackrel{\text { Step } 4}{\hookrightarrow}=36 x^{2}-36 x+36 \newlineFor this reason, Chun stated that the equation is a true identity.\newlineIs Chun correct? If not, in which step did he make a mistake?\newlineChoose 11 answer:\newline(A) Chun is correct.\newline(B) Chun is incorrect. He made a mistake in step 22.\newline(C) Chun is incorrect. He made a mistake in step 33.\newline(D) Chun is incorrect. He made a mistake in step 44.

Full solution

Q. Chun was asked whether the following equation is an identity:\newline3(2x1)2+27=36x236x+36 3(2 x-1)^{2}+27=36 x^{2}-36 x+36 \newlineHe performed the following steps:\newline3(2x1)2+27 3(2 x-1)^{2}+27 \newline Step 1=3(2x1)(2x1)+27 \stackrel{\text { Step } 1}{\hookrightarrow}=3(2 x-1)(2 x-1)+27 \newline Step 2=(6x3)(6x3)+27 \stackrel{\text { Step } 2}{\hookrightarrow}=(6 x-3)(6 x-3)+27 \newline Step 3=36x218x18x+9+27 \stackrel{\text { Step } 3}{\hookrightarrow}=36 x^{2}-18 x-18 x+9+27 \newline Step 4=36x236x+36 \stackrel{\text { Step } 4}{\hookrightarrow}=36 x^{2}-36 x+36 \newlineFor this reason, Chun stated that the equation is a true identity.\newlineIs Chun correct? If not, in which step did he make a mistake?\newlineChoose 11 answer:\newline(A) Chun is correct.\newline(B) Chun is incorrect. He made a mistake in step 22.\newline(C) Chun is incorrect. He made a mistake in step 33.\newline(D) Chun is incorrect. He made a mistake in step 44.
  1. Expand Binomial: Expand the squared binomial. 3(2x1)2+27=3(4x24x+1)+273(2x-1)^{2}+27 = 3(4x^2 - 4x + 1) + 27
  2. Distribute and Add: Distribute the 33 into the binomial and add 2727. \newline= 12x212x+3+2712x^2 - 12x + 3 + 27
  3. Combine Like Terms: Combine like terms.\newline=12x212x+30= 12x^2 - 12x + 30
  4. Compare Equations: Compare the expanded equation to the original equation.\newline12x212x+3036x236x+3612x^2 - 12x + 30 \neq 36x^2 - 36x + 36

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