Final answer:
The given bi-quadratic equation is solved by using substitution to transform it into a quadratic equation, then using the quadratic formula to find the solutions. The solutions for x are ±3, thus the correct answer is option c) {±2, ±9}.
Step-by-step explanation:
To solve the equation x⁴ - 11x² - 18 = 0, we can use the approach of solving quadratic equations. This equation is actually a bi-quadratic equation, which means it is a quadratic in terms of x². Let's substitute y = x², turning the original equation into y² - 11y - 18 = 0. Now it's a regular quadratic equation in terms of y, and we can apply the quadratic formula, which is -b ± √(b² - 4ac) / (2a).
For our equation, a = 1, b = -11, and c = -18. Plugging these values into the quadratic formula, we have y = (-(-11) ± √((-11)² - 4(1)(-18))) / (2(1)). This simplifies to y = (11 ± √(121 + 72)) / 2, which further simplifies to y = (11 ± √193) / 2. This gives us two potential values for y: y = 9 and y = -2.
Remembering that we substituted y = x², we now find the values for x by taking square roots: x = ±√9 and x = ±√-2. Since we cannot have a real square root of a negative number, we ignore the roots of -2. Thus, our real solutions for x are x = ± 3. Therefore, the correct option is c) {±2, ±9}.