C. Quadratic
Step-by-step explanation:
Solve for a:
2 a^3 + a^2 + a + 4 = 0
Hint: | Look for a simple substitution that eliminates the quadratic term of 2 a^3 + a^2 + a + 4.
Eliminate the quadratic term by substituting x = a + 1/6:
23/6 + (x - 1/6)^2 + 2 (x - 1/6)^3 + x = 0
Hint: | Write the cubic polynomial on the left-hand side in standard form.
Expand out terms of the left hand side:
2 x^3 + (5 x)/6 + 104/27 = 0
Hint: | Write the cubic equation in standard form.
Divide both sides by 2:
x^3 + (5 x)/12 + 52/27 = 0
Hint: | Perform the substitution x = y + λ/y.
Change coordinates by substituting x = y + λ/y, where λ is a constant value that will be determined later:
52/27 + 5/12 (y + λ/y) + (y + λ/y)^3 = 0
Hint: | Transform the rational equation into a polynomial equation.
Multiply both sides by y^3 and collect in terms of y:
y^6 + y^4 (3 λ + 5/12) + (52 y^3)/27 + y^2 (3 λ^2 + (5 λ)/12) + λ^3 = 0
Hint: | Find an appropriate value for λ in order to make the coefficients of y^2 and y^4 both zero.
Substitute λ = -5/36 and then z = y^3, yielding a quadratic equation in the variable z:
z^2 + (52 z)/27 - 125/46656 = 0
Hint: | Solve for z.
Find the positive solution to the quadratic equation:
z = 1/216 (3 sqrt(4821) - 208)
Hint: | Perform back substitution on z = 1/216 (3 sqrt(4821) - 208).
Substitute back for z = y^3:
y^3 = 1/216 (3 sqrt(4821) - 208)
Hint: | Take the cube root of both sides.
Taking cube roots gives 1/6 (3 sqrt(4821) - 208)^(1/3) times the third roots of unity:
y = 1/6 (3 sqrt(4821) - 208)^(1/3) or y = -1/6 (208 - 3 sqrt(4821))^(1/3) or y = 1/6 (-1)^(2/3) (3 sqrt(4821) - 208)^(1/3)
Hint: | Perform back substitution with x = y - 5/(36 y).
Substitute each value of y into x = y - 5/(36 y):
x = 1/6 (3 sqrt(4821) - 208)^(1/3) - 5/(6 (3 sqrt(4821) - 208)^(1/3)) or x = -1/6 (208 - 3 sqrt(4821))^(1/3) - (5 (-1)^(2/3))/(6 (3 sqrt(4821) - 208)^(1/3)) or x = 5/6 ((-1)/(3 sqrt(4821) - 208))^(1/3) + 1/6 (-1)^(2/3) (3 sqrt(4821) - 208)^(1/3)
Hint: | Simplify each solution.
Bring each solution to a common denominator and simplify:
x = ((3 sqrt(4821) - 208)^(2/3) - 5)/(6 (3 sqrt(4821) - 208)^(1/3)) or x = -1/6 (208 - 3 sqrt(4821))^(1/3) - (5 (-1)^(2/3))/(6 (3 sqrt(4821) - 208)^(1/3)) or x = 1/6 (5 ((-1)/(3 sqrt(4821) - 208))^(1/3) + (-1)^(2/3) (3 sqrt(4821) - 208)^(1/3))
Hint: | Perform back substitution on the three roots.
Substitute back for a = x - 1/6:
a = ((3 sqrt(4821) - 208)^(2/3) - 5)/(6 (3 sqrt(4821) - 208)^(1/3)) - 1/6 or a = -1/6 - 1/6 (208 - 3 sqrt(4821))^(1/3) - (5 (-1)^(2/3))/(6 (3 sqrt(4821) - 208)^(1/3)) or a = 1/6 (5 (-1/(3 sqrt(4821) - 208))^(1/3) + (-1)^(2/3) (3 sqrt(4821) - 208)^(1/3)) - 1/6
