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If x = [-(q)/(2) + sqrt((q^(2))/(4) + (p...

If `x = [-(q)/(2) + sqrt((q^(2))/(4) + (p^(3))/(27))]^((1)/(3)) + [-(q)/(2) - sqrt((q^(2))/(4) + (p^(3))/(27))]^((1)/(3))` then prove that `x^(3) + pq + q = 0`

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`x = [-(q)/(2) + sqrt((q^(2))/(4) + (p^(3))/(27))]^((1)/(3)) + [-(q)/(2) - sqrt((q^(2))/(4) + (p^(3))/(27))]^((1)/(3))`
or `x^(3) = - (q)/(2) + sqrt((q^(2))/(4) + (p^(3))/(27)) - (q)/(2) - sqrt((q^(2))/(4) + (p^(3))/(27)) - (q)/(2) - sqrt((q^(2))/(4) + (p^(3))/(27)) + 3 [ - (q)/(2) + sqrt((q^(2))/(4) + (p^(3))/(27))]^((1)/(3)) xx [-(q)/(2) - sqrt((q^(2))/(4) + (p^(3))/(27))]^((1)/(3)) [(-(q)/(2) + sqrt((q^(2))/(4) + (p^(3))/(27)))^((1)/(3)) + (-(q)/(2) - sqrt((q^(2))/(4) + (p^(3))/(27)))^((1)/(3))]`
or, `x^(3) = - q + 3 [(-(q)/(2))^(2) - (sqrt((q^(2))/(4) + (p^(3))/(27)))^(2) ]^((1)/(3)) xx (x)`
or, `x^(3) = - q + 3 [(q^(2))/(4) - (q^(2))/(4) - (p^(3))/(27)]^((1)/(3)) xx(x)`
or `x^(3) = -q + 3 [-(p^(3))/(27)]^((1)/(3)) xx x`
or `x^(3) = - q + 3 xx - (p)/(3) xx x or, x^(3) = -q - px`
or, `x^(3) = -q + 3 xx -(p)/(3) xx x or, x^(3) = -q - px`
or, `x^(3) + px + q = 0`
Hence, `x^(3) + px + q = 0`
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CALCUTTA BOOK HOUSE-QUADRATIC SURDS-Exercise -3.2
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  8. (a) If x gt a, y gt b and x + y = a + b, then find which one of (sqrtx...

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  14. If x^((1)/(3)) + y^((1)/(3)) + z^((1)/(3)) = 0, then prove that (x + y...

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