solve for $x$ in the equation $x^2 - 12x + 36 = 90$.$x=6\\pm3\\sqrt{10}$$x=6\\pm2\\sqrt{7}$$x=12\\pm3\\sqrt{2…

Explanation:

Step1: Rearrange equation to standard form

Subtract 80 from both sides: $x^2 - 12x + 36 - 80 = 0$ $x^2 - 12x - 44 = 0$

Step2: Apply quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=1$, $b=-12$, $c=-44$: $x=\frac{12\pm\sqrt{(-12)^2-4(1)(-44)}}{2(1)}$

Step3: Calculate discriminant

Compute value under square root: $\sqrt{144 + 176} = \sqrt{320} = \sqrt{64\times5} = 8\sqrt{5}$ *(Note: Correcting calculation error: $(-12)^2=144$, $-4(1)(-44)=176$, $144+176=320$; $\sqrt{320}=\sqrt{16\times20}=4\sqrt{20}=4\sqrt{4\times5}=8\sqrt{5}$. However, matching options, recheck original equation: $x^2-12x+36=80$ is $(x-6)^2=80$)

Step4: Use perfect square shortcut

Rewrite left side as square: $(x-6)^2=80$ Take square root of both sides: $x-6=\pm\sqrt{80}=\pm4\sqrt{5}$ *(Note: Matching given options, re-express $\sqrt{80}$ as $\sqrt{90-10}$ is incorrect. Correct match: $\sqrt{80}=\sqrt{4\times20}=2\sqrt{20}=2\sqrt{4\times5}=4\sqrt{5}$, but closest option uses $\sqrt{10}$: $(x-6)^2=90-10$ is wrong. Correct derivation for option 1: $(x-6)^2=90$ gives $x=6\pm3\sqrt{10}$. Recheck original equation: $x^2-12x+36=90$ would be correct, so assuming typo $80\to90$) $x=6\pm3\sqrt{10}$

Answer:

$x=6\pm3\sqrt{10}$ (first option)