Solve {c}{frac{x^{2}}{3}+48/x^{2}=10(x/3-4/x}{(x^2+4x+8)^2+3x(x^2+4x+8)+2x^2=0}

$$\left. \begin{array} { c } { \frac { x ^ { 2 } } { 3 } + \frac { 48 } { x ^ { 2 } } = 10 ( \frac { x } { 3 } - \frac { 4 } { x } } \\ { ( x ^ { 2 } + 4 x + 8 ) ^ { 2 } + 3 x ( x ^ { 2 } + 4 x + 8 ) + 2 x ^ { 2 } = 0 } \end{array} \right.$$

Solve for x

$$x=-\sqrt{6}\approx -2.449489743$$

$$x=\sqrt{6}\approx 2.449489743$$

$$x=2\sqrt{6}\approx 4.898979486$$

$$x=-2\sqrt{6}\approx -4.898979486$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x^{2}, the least common multiple of 3,x^{2}.

$$x^{2}x^{2}+3\times 48=30x^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{4}+3\times 48=30x^{2}$$

Multiply 3 and 48 to get 144.

$$x^{4}+144=30x^{2}$$

Subtract 30x^{2} from both sides.

$$x^{4}+144-30x^{2}=0$$

Substitute t for x^{2}.

$$t^{2}-30t+144=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -30 for b, and 144 for c in the quadratic formula.

$$t=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 1\times 144}}{2}$$

Do the calculations.

$$t=\frac{30±18}{2}$$

Solve the equation t=\frac{30±18}{2} when ± is plus and when ± is minus.

$$t=24 t=6$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=2\sqrt{6} x=-2\sqrt{6} x=\sqrt{6} x=-\sqrt{6}$$