Solve (x^2 - 5x) * sqrt(4 + 3x - x^2) = 0 and give the sum of its solutions.

Step-by-step solution

1. 1

   Find the domain before solving anything. The radicand must be non-negative: $4+3x-x^{2}\ge 0$, i.e. $x^{2}-3x-4\le 0$, i.e. $(x-4)(x+1)\le 0$, so $x\in[-1,4].$ Any candidate outside this interval is not a solution of the original equation, no matter what the algebra says.

2. 2

   Apply the zero-product rule. A product is zero exactly when one factor is zero, so either $x^{2}-5x=0$ or $\sqrt{4+3x-x^{2}}=0$.

3. 3

   Solve the polynomial factor and filter by the domain. $x^{2}-5x=x(x-5)=0$ gives $x=0$ or $x=5$. Since $5\notin[-1,4]$, only $x=0$ survives; at $x=5$ the radicand would be $4+15-25=-6<0$.

4. 4

   Solve the radical factor. A square root is zero exactly when its radicand is zero: $4+3x-x^{2}=0$, i.e. $x^{2}-3x-4=0$, i.e. $(x-4)(x+1)=0$, giving $x=4$ and $x=-1$. Both are the endpoints of the domain, so both are admissible.

5. 5

   Collect and add. The solution set is $\{-1,0,4\}$ and the required sum is $-1+0+4=3$. Substituting each value back gives $6\cdot\sqrt{0}$, $0\cdot 2$ and $-4\cdot 0$, all zero, so all three are genuine.