solve for x in the equation $x^{2}-12x + 36 = 90$.\n$x = 6pm3sqrt{10}$\n$x = 6pm2sqrt{7}$\n$x =…

Explanation:

Step1: Recognize the perfect - square trinomial

The left - hand side of the equation (x^{2}-12x + 36) is a perfect - square trinomial. Using the formula ((a - b)^2=a^{2}-2ab + b^{2}), where (a=x) and (b = 6) (since (2ab=2\times x\times6 = 12x)), we can rewrite the equation as ((x - 6)^{2}=90).

Step2: Take the square root of both sides

Taking the square root of both sides, we get (x-6=\pm\sqrt{90}).

Step3: Simplify (\sqrt{90})

Simplify (\sqrt{90}=\sqrt{9\times10}=3\sqrt{10}).

Step4: Solve for (x)

Add (6) to both sides of the equation (x - 6=\pm3\sqrt{10}). So (x=6\pm3\sqrt{10}).

Answer:

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