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(v) (x^(2)-3x-10)(x^(2)-5x-6)=144

2 min read Jun 16, 2024
(v) (x^(2)-3x-10)(x^(2)-5x-6)=144

Solving the Equation: (x^2 - 3x - 10)(x^2 - 5x - 6) = 144

This equation involves a product of two quadratic expressions equaling a constant. To solve it, we'll follow these steps:

  1. Expand the equation: Begin by multiplying the two quadratic expressions on the left-hand side.

    (x^2 - 3x - 10)(x^2 - 5x - 6) = 144

    x^4 - 8x^3 - 11x^2 + 68x + 60 = 144

  2. Move the constant to the left side: Subtract 144 from both sides to obtain a polynomial equation equal to zero.

    x^4 - 8x^3 - 11x^2 + 68x - 84 = 0

  3. Factor the polynomial: Try to factor the polynomial expression. This step can be challenging, but it can be achieved through various methods like grouping or using the Rational Root Theorem.

    For this specific example, the polynomial can be factored as:

    (x - 7)(x + 2)(x - 3)(x + 2) = 0

  4. Solve for x: Now that the expression is factored, set each factor equal to zero and solve for x.

    x - 7 = 0 => x = 7 x + 2 = 0 => x = -2 x - 3 = 0 => x = 3

Therefore, the solutions to the equation (x^2 - 3x - 10)(x^2 - 5x - 6) = 144 are x = 7, x = -2, and x = 3.

Important Note: While factoring is a common method, sometimes it may not be straightforward. In such cases, numerical methods or software tools might be necessary to find the solutions.

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