First, let's distribute the (2-x) in the first part of the equation:
(2-x)(2+x) + 7.5x = 2 + x^2(4 - 2x + 2x - x^2) + 7.5x = 2 + x^2
Now simplify:
4 - x^2 + 7.5x = 2 + x^27.5x - x^2 = -2 + x^2
Rearrange terms:
7.5x = -2 + 2x^22x^2 - 7.5x - 2 = 0
Now, we have a quadratic equation in the form of ax^2 + bx + c = 0. We can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in our values:
x = (7.5 ± sqrt((-7.5)^2 - 42(-2))) / 2*2x = (7.5 ± sqrt(56.25 + 16))/4x = (7.5 ± sqrt(72.25))/4x = (7.5 ± 8.5)/4
So, the possible solutions are:
x = (7.5 + 8.5)/4 = 16/4 = 4x = (7.5 - 8.5)/4 = -1/4 = -0.25
Therefore, the solutions to the equation are x = 4 and x = -0.25.
First, let's distribute the (2-x) in the first part of the equation:
(2-x)(2+x) + 7.5x = 2 + x^2
(4 - 2x + 2x - x^2) + 7.5x = 2 + x^2
Now simplify:
4 - x^2 + 7.5x = 2 + x^2
7.5x - x^2 = -2 + x^2
Rearrange terms:
7.5x = -2 + 2x^2
2x^2 - 7.5x - 2 = 0
Now, we have a quadratic equation in the form of ax^2 + bx + c = 0. We can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in our values:
x = (7.5 ± sqrt((-7.5)^2 - 42(-2))) / 2*2
x = (7.5 ± sqrt(56.25 + 16))/4
x = (7.5 ± sqrt(72.25))/4
x = (7.5 ± 8.5)/4
So, the possible solutions are:
x = (7.5 + 8.5)/4 = 16/4 = 4
x = (7.5 - 8.5)/4 = -1/4 = -0.25
Therefore, the solutions to the equation are x = 4 and x = -0.25.