Question

Solve for x.


(x-7)(x-3)(x+5)(x+1)= 1680

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Answer 1

Step-by-step explanation:

Given Equation is (x - 7)(x - 3)(x + 5)(x + 1) = 1680

⇒ (x² - 10x + 21)(x + 5)(x + 1) = 1680

⇒ (x³ - 5x² - 29x + 105)(x + 1) = 1680

⇒ x⁴ - 4x³ - 34x² + 76x + 105 = 1680

⇒ x⁴ - 4x³ - 34x² + 76x - 1575 = 0

⇒ x⁴ - 11x³ + 43x² - 225x + 7x³ - 77x² + 301x - 1575 = 0

⇒ x(x³ - 11x² + 43x - 225) + 7(x³ - 11x² + 43x - 225) = 0

⇒ (x + 7)(x³ - 11x² + 43x - 225) = 0

⇒ (x + 7)[x³ - 2x² + 25x - 9x² + 18x - 225] = 0

⇒ (x + 7)[x(x² - 2x + 25) - 9(x² - 2x + 25)] = 0

⇒ (x + 7)(x - 9)(x² - 2x + 25) = 0

(i)

x + 7 = 0

x = -7

(ii)

x + 9 = 0

x = -9

(iii)

x² - 2x + 25 = 0

∴ x = (-b ± √b² - 4ac)/2a

     = (2 ± √96 i)/2

     = 1 ± 2√6 i

   

∴ x = -7, 9, 1 ± 2√6 i

Hope it helps!

Answer 2

Given Equation is (x - 7)(x - 3)(x + 5)(x + 1) = 1680

(x^2 - 10x + 21)(x + 5)(x + 1) = 1680

(x^3 - 5x^2 - 29x + 105)(x + 1) = 1680

x^4 - 4x^3 - 34x^2 + 76x + 105 = 1680

x^4 - 4x^3 - 34x^2 + 76x - 1575 = 0

x^4 - 11x^3 + 43x^2 - 225x + 7x^3 - 77x^2 + 301x - 1575 = 0

x(x^3 - 11x^2 + 43x - 225) + 7(x^3 - 11x^2 + 43x - 225) = 0

(x + 7)(x^3 - 11x^2 + 43x - 225) = 0

(x + 7)[x^3 - 2x^2 + 25x - 9x^2 + 18x - 225] = 0

(x + 7)[x(x^2 - 2x + 25) - 9(x^2 - 2x + 25)] = 0

(x + 7)(x - 9)(x^2 - 2x + 25) = 0

(1)

x + 7 = 0

x = -7

(2)

x + 9 = 0

x = -9

(iii)

(x^2 - 2x + 25) = 0

x = (-b ± √b² - 4ac)/2a

     = (2 ± √96 i)/2

     = 1 ± 2√6 i

So,solutions are -7,-9, 1 ± 2√6 i

Hope it will helps you