What is X^3-2X^2+5/4X^2-3

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Mathematics

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31 May, 01:49

What is X^3-2X^2+5/4X^2-3

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Leroy Steele · Answer 1

X = ((56 sqrt (3) + 97) ^ (2/3) + 1) / (4 (56 sqrt (3) + 97) ^ (1/3)) + 1/4 or X = ((-1) ^ (2/3) - (-1) ^ (1/3) (56 sqrt (3) + 97) ^ (2/3)) / (4 (56 sqrt (3) + 97) ^ (1/3)) + 1/4 or X = 1/4 ((-1) ^ (2/3) (97 + 56 sqrt (3)) ^ (1/3) - (56 sqrt (3) - 97) ^ (1/3)) + 1/4

Step-by-step explanation:

Solve for X:

X^3 - (3 X^2) / 4 - 3 = 0

Bring X^3 - (3 X^2) / 4 - 3 together using the common denominator 4:

1/4 (4 X^3 - 3 X^2 - 12) = 0

Multiply both sides by 4:

4 X^3 - 3 X^2 - 12 = 0

Eliminate the quadratic term by substituting x = X - 1/4:

-12 - 3 (x + 1/4) ^2 + 4 (x + 1/4) ^3 = 0

Expand out terms of the left hand side:

4 x^3 - (3 x) / 4 - 97/8 = 0

Divide both sides by 4:

x^3 - (3 x) / 16 - 97/32 = 0

Change coordinates by substituting x = y + λ/y, where λ is a constant value that will be determined later:

-97/32 - 3/16 (y + λ/y) + (y + λ/y) ^3 = 0

Multiply both sides by y^3 and collect in terms of y:

y^6 + y^4 (3 λ - 3/16) - (97 y^3) / 32 + y^2 (3 λ^2 - (3 λ) / 16) + λ^3 = 0

Substitute λ = 1/16 and then z = y^3, yielding a quadratic equation in the variable z:

z^2 - (97 z) / 32 + 1/4096 = 0

Find the positive solution to the quadratic equation:

z = 1/64 (97 + 56 sqrt (3))

Substitute back for z = y^3:

y^3 = 1/64 (97 + 56 sqrt (3))

Taking cube roots gives 1/4 (97 + 56 sqrt (3)) ^ (1/3) times the third roots of unity:

y = 1/4 (97 + 56 sqrt (3)) ^ (1/3) or y = - 1/4 (-97 - 56 sqrt (3)) ^ (1/3) or y = 1/4 (-1) ^ (2/3) (97 + 56 sqrt (3)) ^ (1/3)

Substitute each value of y into x = y + 1 / (16 y):

x = 1 / (4 (56 sqrt (3) + 97) ^ (1/3)) + 1/4 (56 sqrt (3) + 97) ^ (1/3) or x = (-1) ^ (2/3) / (4 (56 sqrt (3) + 97) ^ (1/3)) - 1/4 (-56 sqrt (3) - 97) ^ (1/3) or x = 1/4 (-1) ^ (2/3) (56 sqrt (3) + 97) ^ (1/3) - 1/4 ((-1) / (56 sqrt (3) + 97)) ^ (1/3)

Bring each solution to a common denominator and simplify:

x = ((56 sqrt (3) + 97) ^ (2/3) + 1) / (4 (97 + 56 sqrt (3)) ^ (1/3)) or x = ((-1) ^ (2/3) - (-1) ^ (1/3) (56 sqrt (3) + 97) ^ (2/3)) / (4 (97 + 56 sqrt (3)) ^ (1/3)) or x = 1/4 ((-1) ^ (2/3) (56 sqrt (3) + 97) ^ (1/3) - (56 sqrt (3) - 97) ^ (1/3))

Substitute back for X = x + 1/4:

Answer: X = ((56 sqrt (3) + 97) ^ (2/3) + 1) / (4 (56 sqrt (3) + 97) ^ (1/3)) + 1/4 or X = ((-1) ^ (2/3) - (-1) ^ (1/3) (56 sqrt (3) + 97) ^ (2/3)) / (4 (56 sqrt (3) + 97) ^ (1/3)) + 1/4 or X = 1/4 ((-1) ^ (2/3) (97 + 56 sqrt (3)) ^ (1/3) - (56 sqrt (3) - 97) ^ (1/3)) + 1/4