Question

(x-2)(x-10)(x-4)(x-8)=64

Answer 1

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Answer 2

To solve the equation (x-2)(x-10)(x-4)(x-8) = 64, we can start by expanding the left side of the equation and then simplify it. Let's proceed with the steps:

(x-2)(x-10)(x-4)(x-8) = 64

Expanding the left side:

(x^2 - 2x - 10x + 20)(x-4)(x-8) = 64

(x^2 - 12x + 20)(x-4)(x-8) = 64

Now, let's multiply (x-4)(x-8):

(x^2 - 12x + 20)(x^2 - 12x + 32) = 64

Expanding the above expression:

x^4 - 12x^3 + 32x^2 - 12x^3 + 144x^2 - 384x + 20x^2 - 240x + 640 = 64

Combining like terms:

x^4 - 24x^3 + 196x^2 - 624x + 640 = 64

Rearranging the equation and simplifying:

x^4 - 24x^3 + 196x^2 - 624x + 576 = 0

Now, we have a quartic equation. Unfortunately, there is no general formula to solve quartic equations. However, we can attempt to factorize or use numerical methods to find the solutions.

By observing the equation, we can try substituting different values for x and check if they satisfy the equation. One possible solution is x = 6:

6^4 - 24(6)^3 + 196(6)^2 - 624(6) + 576 = 0

1296 - 10368 + 7056 - 3744 + 576 = 0

0 = 0

Since the equation is satisfied, x = 6 is a solution to the equation.

Using numerical methods or factoring techniques, we can find additional solutions if they exist. However, the process can be complex, and the equation may have non-real or irrational solutions.