(b) Given one of the zeros of equation 2x³ - 23x + 78x +k=0 is 3/2 Find value k, hence,
solve the equation.

k = -55 x - 2 x^3

Step-by-step explanation:

Solve for k:

2 x^3 + 55 x + k = 0

Hint: | Solve for k.

Subtract 2 x^3 + 55 x from both sides:

k = -55 x - 2 x^3

3/2 is not a Root of this equation - see below, those are the Roots/Zeros.

Solve for x:

2 x^3 + 55 x + k = 0

Divide both sides by 2:

x^3 + (55 x)/2 + k/2 = 0

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

k/2 + 55/2 (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 + 55/2) + (k y^3)/2 + y^2 (3 λ^2 + (55 λ)/2) + λ^3 = 0

Substitute λ = -55/6 and then z = y^3, yielding a quadratic equation in the variable z:

z^2 + (k z)/2 - 166375/216 = 0

Find the positive solution to the quadratic equation:

z = 1/36 (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)

Substitute back for z = y^3:

y^3 = 1/36 (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)

Taking cube roots gives (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3)/6^(2/3) times the third roots of unity:

y = (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3)/6^(2/3) or y = -((-1)^(1/3) (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3))/6^(2/3) or y = ((-1)^(2/3) (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3))/6^(2/3)

Substitute each value of y into x = y - 55/(6 y):

x = (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3)/6^(2/3) - 55/(6^(1/3) (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3)) or x = -(55 (-1)^(2/3))/(6^(1/3) (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3)) - ((-1)^(1/3) (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3))/6^(2/3) or x = (55 ((-1)/6)^(1/3))/(sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3) + ((-1)^(2/3) (sqrt(3) sqrt(27 k^2 + 332750) - 9 k)^(1/3))/6^(2/3)

Bring each solution to a common denominator and simplify:

x = (6^(1/3) (sqrt(81 k^2 + 998250) - 9 k)^(2/3) - 55 6^(2/3))/(6 (sqrt(81 k^2 + 998250) - 9 k)^(1/3)) or x = -((-1)^(1/3) ((sqrt(81 k^2 + 998250) - 9 k)^(2/3) + 55 (-6)^(1/3)))/(6^(2/3) (sqrt(81 k^2 + 998250) - 9 k)^(1/3)) or x = ((-1)^(1/3) ((-1)^(1/3) (sqrt(81 k^2 + 998250) - 9 k)^(2/3) + 55 6^(1/3)))/(6^(2/3) (sqrt(81 k^2 + 998250) - 9 k)^(1/3))

lockers 1, 4, 9, 16, 25, 40, and 49 are open

i ran through it by the other answer is wrong

16 is the absolute value

step-by-step explanation: