Find all solutions of the equation and express them in the form a+bi

2x^2+3=2x

2x^2+3x=2x is the same as rewriting it as 2x^2-2x+3=0

The zeros using the quadratic formula. A=1 B= -2 C= 3
I ended up with

x=1/2(1+ i√5) also written as 1/2 + i√5/2
Both will give you the same answer

The other solution is

x=1/2(1-i√5) or 1/2-i√5/2

IMPORTANT: when writing the i√5/2 part the 2 is NOT under the radical with the 5 it is the entire component over 2

Well, well, well... looks like this equation wants to be the center of attention! Let's give it the spotlight and solve it:

To begin, let's rearrange the equation and bring all the terms to one side:

2x^2 - 2x + 3 = 0

Now, we can use the quadratic formula to find the solutions. The quadratic formula goes like this:

x = (-b ± √(b^2 - 4ac)) / (2a)

Alright, let's identify the coefficients:
a = 2, b = -2, and c = 3.

Now, plug the values into the formula, and let's see what happens:

x = (-(-2) ± √((-2)^2 - 4(2)(3))) / (2(2))
x = (2 ± √(4 - 24)) / 4
x = (2 ± √(-20)) / 4

Uh-oh! Looks like we've ended up with a negative square root. This means the solutions will involve imaginary numbers. Let's simplify it further:

x = (2 ± √(-1)(4)) / 4
x = (2 ± 2i√5) / 4
x = (1 ± i√5) / 2

There you have it! The solutions to the equation 2x^2 + 3 = 2x expressed in the form a + bi are:

x = 1/2 + i√5/2
x = 1/2 - i√5/2

Keep in mind that "i" represents the imaginary unit, which is the square root of -1. So, the clownish solutions are all set and ready to entertain you!

To find the solutions of the equation 2x^2 + 3 = 2x, we need to rearrange the equation to obtain a quadratic expression in standard form (ax^2 + bx + c = 0).

Step 1: Move all terms to one side of the equation:
2x^2 - 2x + 3 = 0

Step 2: Divide the entire equation by the leading coefficient (2) to simplify the equation:
x^2 - x + (3/2) = 0

Now, we can solve the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -1, and c = 3/2.

Step 3: Substitute the values into the quadratic formula and simplify:

x = (-( -1) ± √((-1)^2 - 4(1)(3/2))) / (2(1))
x = (1 ± √(1 - 6)) / 2
x = (1 ± √(-5)) / 2

Since the discriminant (√(1 - 4(1)(3/2) = √(1 - 6) = √(-5)) is a negative number, the solutions will involve complex numbers.

Step 4: Simplify further by expressing the square root of -5 in terms of "i" (imaginary unit):
x = (1 ± √5i) / 2

This gives us the two solutions in the form a + bi:
x = (1 + √5i) / 2
x = (1 - √5i) / 2

Therefore, the solutions to the equation 2x^2 + 3 = 2x are:
x = (1 + √5i) / 2
x = (1 - √5i) / 2

To find all the solutions of the equation 2x^2 + 3 = 2x and express them in the form a + bi, we need to solve the equation for x.

Step 1: Move all terms to one side to set the equation equal to zero:
2x^2 - 2x + 3 = 0

Step 2: Use the quadratic formula to solve the quadratic equation: ax^2 + bx + c = 0.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac))/ (2a).

For our equation, a = 2, b = -2, and c = 3. Substituting these values into the quadratic formula, we get:

x = (-(-2) ± √((-2)^2 - 4(2)(3))) / (2(2))
x = (2 ± √(4 - 24)) / 4
x = (2 ± √(-20)) / 4

Step 3: Simplify the square root of -20:
√(-20) = √(-1 * 4 * 5) = 2i√5

Step 4: Substitute the simplified value into the equation:
x = (2 ± 2i√5) / 4

Step 5: Simplify the expression:
x = (1 ± i√5) / 2

Hence, the solutions of the equation 2x^2 + 3 = 2x, expressed in the form a + bi, are:
x = (1 + i√5)/2 and x = (1 - i√5)/2.