ix^2-2x+i=0
-x^2 -2xi -1 = 0
x^2 + 2i x +1 = 0
x = [ -2i +/- sqrt ( -4 - 4) ]/2
x = [ -2i +/- 2 i sqrt 2 ] /2
x = 2i (-1 +/- sqrt 2 ) /2
x = (-1 +/- sqrt 2) i
To solve the quadratic equation ix^2 - 2x + i = 0, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = i, b = -2, and c = i. Substituting these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(i)(i))) / (2(i))
x = (2 ± √(4 + 4)) / (2i)
x = (2 ± √8) / (2i)
Simplifying further, we have:
x = (2 ± 2√2) / (2i)
Now, we can simplify the expression by dividing both the numerator and denominator by 2:
x = (1 ± √2) / i
To eliminate the imaginary denominator, we can multiply both the numerator and denominator by -i:
x = [(1 ± √2) / i] * [-i / -i]
x = (1 ± √2)*(-i) / [i*(-i)]
x = (-1 ± √2) / (i^2)
Since i^2 is defined to be -1, we have:
x = (-1 ± √2) / (-1)
Finally, simplifying the expression, we get the solutions:
x = 1 ± √2
To solve the quadratic equation, ix^2 - 2x + i = 0, we can use the quadratic formula. The quadratic formula states that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, since the coefficient of x^2 is ix^2, we can rewrite the equation as:
(-2x ± √((-2)^2 - 4(i)(i))) / (2i)
Simplifying further:
(-2x ± √(4 + 4)) / (2i)
(-2x ± √8) / (2i)
Now, we can simplify the expression √8 by breaking it down into √(4 * 2):
(-2x ± 2√2) / (2i)
Next, we can simplify the expression (-2x ± 2√2) / (2i) further by canceling out the common factor of 2 in the numerator and denominator:
(-x ± √2) / i
Multiplying the numerator and denominator by (-i) will help us change i to -i:
((-x ± √2) / i) * (-i / -i)
This gives us:
(x ± √2i) / i
Simplifying, we get:
x ± √2i/i
To divide a complex number by i, we can multiply both the numerator and denominator by the conjugate of i, which is -i:
(x ± √2i/i) * (i/-i)
Multiplying, we have:
(x ± i√2) / -1
Finally, we can simplify further by multiplying the numerator by -1:
-(x ± i√2)
Therefore, the solutions for the quadratic equation ix^2 - 2x + i = 0 are:
x = -(x + i√2)
x = -(x - i√2)