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Without solving, determine the character of the solutions of each equation in the complex number system.
1. 2x^2 - 4x + 1 = 0
2. x^2 + 6 = 2x
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(1) For this quadratic equation, calculate the discriminant
d = b^2 - 4ac.
Referring to the general form equation, a= 2; b= -4; c= 1; so
d = (-4)^2 - 4*2*1 = 16 - 8 = 8.
The discriminant is positive; so this equation has two distinct real solutions.
At this point, the analysis for part (1) is complete.
(2) Write the given quadratic equation in standard form
x^2 - 2x + 6 = 0.
For this quadratic equation, calculate the discriminant
d = b^2 - 4ac.
In this case, a= 1; b= -2; c= 6; so
d = (-2)^2 - 4*1*6 = 4 - 24 = -20.
The discriminant is negative; so this equation has no real solutions.
It has two distinct solutions in complex numbers.
At this point, the analysis for part (2) is complete.
Solved, in full.