2230.41 – Find the Smallest Root


Find the smallest root of the equation:

(x+5)(x+3)(x+5)(2x+7)=0(x+5)(x+3) - (x+5)(2x+7) = 0


Solution

0=(x+5)(x+3)(x+5)(2x+7)=(x+5)(x+3(2x+7))=(x+5)(x4)=(x+5)(x+4)\begin{aligned}0 &= (x+5)(x+3) - (x+5)(2x+7) \\&= (x+5)\left(x+3 - (2x+7)\right) \\&= (x+5)(-x-4) \\&= (x+5)(x + 4)\end{aligned}

Either x=4x = -4 or x=5x = -5. The smallest root is 5-5.