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(x^2-1)(x^2-4)(x^2-7)(x^2-10) 0

2 min read Jun 17, 2024
(x^2-1)(x^2-4)(x^2-7)(x^2-10) 0

Solving the Equation (x^2-1)(x^2-4)(x^2-7)(x^2-10) = 0

This equation involves a product of four quadratic expressions, each of which can be factored further. To solve for the values of 'x' that satisfy this equation, we can utilize the Zero Product Property. This property states that if the product of several factors is equal to zero, then at least one of the factors must be equal to zero.

Applying the Zero Product Property

  1. Factor the quadratic expressions:

    • (x^2 - 1) = (x + 1)(x - 1)
    • (x^2 - 4) = (x + 2)(x - 2)
    • (x^2 - 7) = (x + √7)(x - √7)
    • (x^2 - 10) = (x + √10)(x - √10)

  2. Rewrite the equation with the factored expressions: (x + 1)(x - 1)(x + 2)(x - 2)(x + √7)(x - √7)(x + √10)(x - √10) = 0

  3. Set each factor equal to zero:

    • x + 1 = 0
    • x - 1 = 0
    • x + 2 = 0
    • x - 2 = 0
    • x + √7 = 0
    • x - √7 = 0
    • x + √10 = 0
    • x - √10 = 0

  4. Solve for 'x' in each equation:

    • x = -1
    • x = 1
    • x = -2
    • x = 2
    • x = -√7
    • x = √7
    • x = -√10
    • x = √10

Solution

Therefore, the solutions to the equation (x^2-1)(x^2-4)(x^2-7)(x^2-10) = 0 are:

x = -1, 1, -2, 2, -√7, √7, -√10, √10

These are the eight distinct values of 'x' that make the product of the factors equal to zero.

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