X2-4/8- 2x+3/3=-1

To solve this equation, we need to find the values of x that satisfy the equation.

First, let's simplify the equation to make it easier to work with.

The given equation is:

x^2 - 4/8 - 2x + 3/3 = -1

We can simplify the equation by finding a common denominator for the fractions:

x^2 - 1/2 - 2x + 1 = -1

Now, we can combine like terms:

x^2 - 2x - 1/2 = -1

To remove the fraction, we can multiply the entire equation by 2:

2(x^2 - 2x - 1/2) = 2(-1)

Expanding this equation:

2x^2 - 4x - 1 = -2

Rearranging the equation:

2x^2 - 4x - 1 + 2 = 0

2x^2 - 4x + 1 = 0

Now, we can attempt to solve this quadratic equation. We can either factor it or use the quadratic formula.

Factoring method:

The quadratic equation does not appear to factor nicely, so let's use the quadratic formula:

The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our equation, a = 2, b = -4 and c = 1:

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

x = (4 ± √(16 - 8)) / 4

x = (4 ± √8) / 4

Simplifying the square root:

x = (4 ± √(4 * 2)) / 4

x = (4 ± 2√2) / 4

We can simplify further by factoring out the common factor of 2:

x = 2(2 ± √2) / 4

x = (2 ± √2) / 2

So, the solutions for the given equation are:

x = (2 + √2) / 2 and x = (2 - √2) / 2