Solve the following equation by using the quadratic formula. Please show all your work to receive full credit.
`-3x^{2}+10x-3=0`
To solve the quadratic equation `-3x^2 + 10x - 3 = 0` using the quadratic formula, let's first identify the values of `a`, `b`, and `c` in the general quadratic equation form `ax^2 + bx + c = 0`.
In this case:
`a = -3`
`b = 10`
`c = -3`
The quadratic formula states that for any quadratic equation `ax^2 + bx + c = 0`, the solutions for `x` can be found using the formula:
`x = (-b ± √(b^2 - 4ac)) / (2a)`
Substituting the values, we get:
`x = (-(10) ± √((10)^2 - 4(-3)(-3))) / (2(-3))`
Simplifying the formula further:
`x = (-10 ± √(100 - 36)) / (-6)`
`x = (-10 ± √(64)) / (-6)`
`x = (-10 ± 8) / (-6)`
Now, we can find the two possible values of `x` by considering both the positive and negative square root:
For the positive square root:
`x = (-10 + 8) / (-6) = -2 / (-6) = 1/3`
For the negative square root:
`x = (-10 - 8) / (-6) = -18 / (-6) = 3`
Therefore, the solutions to the equation `-3x^2 + 10x - 3 = 0` are `x = 1/3` and `x = 3`.