Question

Find The Sum Of All Solutions To The Equation `3x(x+4)=66`. Help, Please?

Answer 1

`-4`

Explanation:

Start with `3x(x+4)=66`

divide through by 3:

`x(x+4)=22`

expand the left side:

`x^2+4x=22`

subtract 22 from both sides:

`x^2+4x-22=0`.

Use the quadratic formula: `x=(-b\pmsqrt(b^2-4ac))/(2a)`:

`x=(-4+sqrt(4^2-4(1)(-22)))/(2)=-2+sqrt(26)` or `x=(-4-sqrt(4^2-4(1)(-22)))/(2)=-2-sqrt(26)`

If we add those together the radicals cancel out:

`-2+sqrt(26) +(-2-sqrt(26)) = -4`, which is just actually the opposite of b/a: sum of roots is `-b/a`, but most of my students never remember that so they go through all the work I just did.