Solve the equation for all real solutions in simplest form.q, squared, minus, 7, q, plus, 3, equals, minus, 2, qq 2 −7q+3=−2q

The given equation is q^2 - 7q + 3 = -2q.

First, let's simplify the equation by moving all terms to one side.

q^2 - 7q + 2q + 3 = 0

This simplifies to:

q^2 - 5q + 3 = 0

Next, we solve for q by factoring the quadratic equation.

Unfortunately, this equation does not factor nicely, so we will have to use the quadratic formula, which is q = [-b ± sqrt(b^2 - 4ac)] / (2a).

In this equation, a = 1, b = -5, and c = 3.

Substituting these values into the quadratic formula gives:

q = [5 ± sqrt((-5)^2 - 413)] / (2*1)

Solving inside the square root first:

q = [5 ± sqrt(25 - 12)] / 2

This simplifies to:

q = [5 ± sqrt(13)] / 2

So, the solutions to the equation are q = (5 + sqrt(13)) / 2 and q = (5 - sqrt(13)) / 2.