What is the answer to 2z^2-7z+4=i(2z+1) ?

The answer to 2z^2-7z+4=i(2z+1) is z=0.48415…-0.38873…i,z=3.01584…+1.38873…i

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2z^2-7z+4=i(2z+1)

2 z^{2} - 7 z + 4 = i (2 z + 1)

z = 0.48415 \dots - 0.38873 \dots i, z = 3.01584 \dots + 1.38873 \dots i

Solution steps

2 z^{2} - 7 z + 4 = i (2 z + 1)

Substitute

z = x + y i

2 (x + y i)^{2} - 7 (x + y i) + 4 = i (2 (x + y i) + 1)

Expand

(2 x^{2} - 2 y^{2} - 7 x + 4) + i (- 7 y + 4 x y) = - 2 y + i (2 x + 1)

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[2 x^{2} - 2 y^{2} - 7 x + 4 = - 2 y - 7 y + 4 x y = 2 x + 1]

[2 x^{2} - 2 y^{2} - 7 x + 4 = - 2 y - 7 y + 4 x y = 2 x + 1] : (x = 0.48415 \dots, x = 3.01584 \dots, y = - 0.38873 \dots y = 1.38873 \dots)

Substitute back

z = x + y i

z = 0.48415 \dots - 0.38873 \dots i, z = 3.01584 \dots + 1.38873 \dots i

x + 2 + 7 i = y i - 2

i (x) = - x^{2} + 80 x + 650

(1 + i) (x + 2 y) + (3 - 2 i) (x - y) = 8 + 3 i 3 i

3 x + (y - x) i = 6

z^{2} + 2 - 23 i = 0

What is the answer to 2z^2-7z+4=i(2z+1) ?
The answer to 2z^2-7z+4=i(2z+1) is z=0.48415…-0.38873…i,z=3.01584…+1.38873…i