Help how would I solve the quadratic equation by completing the square: 2z^2 - 5z + 1 = 0

2 [ z ² - (5/2) x ] = - 1

[ z ² - (5/2) x ] = - 1/2

z ² - (5/2) x + 25/16 = - 1/2 + 25/16

[ z - (5/4) ] ² = 17/16

z - 5/4 = ± √17 / 4

z = 5/4 ± √17 / 4

Thank you ---pleased to help.

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let's first write: 2z^2 - 5z + 1 = 2(z^2 - 5z/2 + 1/2)
Now look at the term in z, its coefficient is -5/2;
divide it by 2, and then square it, to obtain (5/4)^2 = 25/16.
This is to obtain z^2 - 5z/2 + 25/16 = (z - 5/4)^2

So we must add 25/16 to our original expression, but also subtract it again to balance out:
2(z^2 - 5z/2 + 1/2) = 2(z^2 - 5z/2 + 25/16 - 25/16 + 1/2)
= 2(z^2 - 5z/2 + 25/16) + 2(1/2 - 25/16)
= 2(z - 5/4)^2 - 17/8

So:
0 = 2z^2 - 5z + 1 = 2(z - 5/4)^2 - 17/8, or:
(z - 5/4)^2 = 17/16

Now you take the square root (with two possible values to it, positive and negative), and add 5/4 to get z.