1. ## Polynomial Equations help

Find the sum of the solutions of the equation

(3x - 8/x)^2 - 7(3x - 8/x)^2 +10 = 0

2. Hi there sri340

Notice there are some like terms here

$\displaystyle \left( 3x-\frac{8}{x}\right)^2-7\left( 3x-\frac{8}{x}\right)^2+10 = 0$

$\displaystyle -6\left( 3x-\frac{8}{x}\right)^2+10 = 0$

$\displaystyle 6\left( 3x-\frac{8}{x}\right)^2 = 10$

Now expand the brackets and apply the quadratic fomula to solve for $\displaystyle x$

3. Hello, sri340!

I suspect there is a typo . . .

Find the sum of the solutions of the equation: .$\displaystyle \left(3x - \frac{8}{x}\right)^2 - 7\left(3x - \frac{8}{x}\right) +10 \:=\: 0$

Expand: .$\displaystyle 9x^2 - 48 + \frac{64}{x^2} - 21x + \frac{56}{x} + 10 \:=\:0 \quad\Rightarrow\quad 9x^2 -21x - 38 + \frac{56}{x} + \frac{64}{x^2} \:=\:0$

Multiply by $\displaystyle x^2\!:\quad 9x^4 - 21x^3 - 38x^2 + 56x + 64 \:=\:0$

Divide by 9: . $\displaystyle x^4 - \frac{7}{3}x^3 - \frac{38}{9}x^2 + \frac{56}{9}x + \frac{64}{9} \:=\:0$

The sum of the roots is the negative of the $\displaystyle x^3$ coefficient: .$\displaystyle \frac{7}{3}$