Sure, let's solve the given equation, u^2 + 10u + 25 = 0. This is a quadratic equation and it is in the standard form: au^2 + bu + c = 0. In this case, a is 1, b is 10, and c is 25.
A quadratic equation can have a maximum of two solutions (real or complex). The solutions for this equation can be found by using the quadratic formula u = [-b ± sqrt(b^2 - 4ac)] / 2a.
Let's calculate the discriminant first, which is the term inside the square root in the quadratic formula. The discriminant is given by: b^2 - 4ac.
Substituting a = 1, b = 10, c = 25, we get discriminant = (10)^2 - 4* 1 * 25 = 100 - 100 = 0.
When the discriminant equals zero, it means the quadratic equation has exactly "one distinct" real solution. This distinct real solution is u = [ -10 ± sqrt(0) ] / 2*1 = -5.
Therefore, the given equation, u^2 + 10u + 25 = 0, has exactly one real solution and it is -5.