2a^3 + a^2 + a + 4
A Linear
B Cubic
C Quadratic
D: Quartic

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

more detail

step-by-step explanation:

false

step-by-step explanation:

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there 2 different things