Final answer:
To solve the quadratic inequality x^2 - 36 > 12x, rearrange it to x^2 - 12x - 36 > 0, factor to (x-18)(x+6) > 0, and determine that the solution is x < -6 or x > 18.
Step-by-step explanation:
To solve the quadratic inequality x² - 36 > 12x, we first need to rearrange the inequality to standard form. This involves moving all terms to one side to get x² - 12x - 36 > 0. The next step is to factor the quadratic expression, if possible, or use the quadratic formula to find the critical points (the values of x where the inequality is equal to zero).
For simplicity, we'll try to factor the expression. We are looking for two numbers that multiply to -36 and add up to -12, which are -18 and 6. Thus, the factored form is (x-18)(x+6) > 0. Now, to determine the solution, we consider the sign changes of the expression over the number line divided by its zeroes, x = 18 and x = -6.
We test intervals to decide where the inequality holds true: Values less than -6, values between -6 and 18, and values greater than 18. We find that the solution for the inequality x² - 12x - 36 > 0 exists where the product of the factors is positive; that is, x < -6 or x > 18.