Final answer:
To solve the given equations for x, follow the steps provided for each equation. For equation (a), isolate x by subtracting 14i from both sides and dividing by 2. For equation (b), factor out x(x-i) and solve for x. For equation (c), use the quadratic formula to find the solutions. And for equation (d), rearrange it and use the quadratic formula to find the solutions.
Step-by-step explanation:
To solve the equation 2x + 14i = 1 for x, we need to isolate the variable x. Subtracting 14i from both sides gives us 2x = 1 - 14i. Then, dividing both sides by 2, we have x = (1 - 14i) / 2. This gives us the value of x in terms of a complex number.
For the equation x² - ix = 0, we can factor out an x from the expression to get x(x - i) = 0. Setting each factor equal to zero, we have x = 0 and x - i = 0. Solving for x in the second equation gives us x = i.
For the equation x² + 2ix - 1 = 0, we can use the quadratic formula to find the values of x. Plugging in the values a = 1, b = 2i, and c = -1 into the formula, we get x = (-2i ± √(4i² - 4(-1))) / 2. Simplifying further, we have x = (-2i ± √(4 - 4)) / 2, which simplifies to x = -i ± √0. Since the square root of 0 is 0, we have x = -i.
Lastly, for the equation ix² - 2x + i = 0, we can rearrange it as x² - (2i/i)x - i = 0. Simplifying further, x² - 2ix - i = 0. Using the quadratic formula with a = 1, b = -2i, and c = -i, we get x = (2i ± √(4i² + 4i)) / 2. Simplifying further, we have x = i ± √(-4 - 4i) / 2. The square root of -4 can be written as 2i√1, so we have x = i ± 2i√1 - i / 2, which simplifies to x = 2i - i / 2 or x = -2i - i / 2. Simplifying even further, we have x = 3i / 2 or x = -3i / 2.