Solve 5/w^2-32=6/w^2+56

$$\frac { 5 } { w ^ { 2 } } - 32 = \frac { 6 } { w ^ { 2 } } + 56$$

Solve for w

$$w=-\frac{\sqrt{22}i}{44}\approx -0-0.106600358i$$

$$w=\frac{\sqrt{22}i}{44}\approx 0.106600358i$$

Steps by Finding Square Root

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w^{2}.

$$5+w^{2}\left(-32\right)=6+w^{2}\times 56$$

Subtract w^{2}\times 56 from both sides.

$$5+w^{2}\left(-32\right)-w^{2}\times 56=6$$

Combine w^{2}\left(-32\right) and -w^{2}\times 56 to get -88w^{2}.

$$5-88w^{2}=6$$

Subtract 5 from both sides.

$$-88w^{2}=6-5$$

Subtract 5 from 6 to get 1.

$$-88w^{2}=1$$

Divide both sides by -88.

$$w^{2}=-\frac{1}{88}$$

The equation is now solved.

$$w=\frac{\sqrt{22}i}{44} w=-\frac{\sqrt{22}i}{44}$$

Steps Using the Quadratic Formula

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w^{2}.

$$5+w^{2}\left(-32\right)=6+w^{2}\times 56$$

Subtract 6 from both sides.

$$5+w^{2}\left(-32\right)-6=w^{2}\times 56$$

Subtract 6 from 5 to get -1.

$$-1+w^{2}\left(-32\right)=w^{2}\times 56$$

Subtract w^{2}\times 56 from both sides.

$$-1+w^{2}\left(-32\right)-w^{2}\times 56=0$$

Combine w^{2}\left(-32\right) and -w^{2}\times 56 to get -88w^{2}.

$$-1-88w^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-88w^{2}-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -88 for a, 0 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{0±\sqrt{0^{2}-4\left(-88\right)\left(-1\right)}}{2\left(-88\right)}$$

Square 0.

$$w=\frac{0±\sqrt{-4\left(-88\right)\left(-1\right)}}{2\left(-88\right)}$$

Multiply -4 times -88.

$$w=\frac{0±\sqrt{352\left(-1\right)}}{2\left(-88\right)}$$

Multiply 352 times -1.

$$w=\frac{0±\sqrt{-352}}{2\left(-88\right)}$$

Take the square root of -352.

$$w=\frac{0±4\sqrt{22}i}{2\left(-88\right)}$$

Multiply 2 times -88.

$$w=\frac{0±4\sqrt{22}i}{-176}$$

Now solve the equation w=\frac{0±4\sqrt{22}i}{-176} when ± is plus.

$$w=-\frac{\sqrt{22}i}{44}$$

Now solve the equation w=\frac{0±4\sqrt{22}i}{-176} when ± is minus.

$$w=\frac{\sqrt{22}i}{44}$$

The equation is now solved.

$$w=-\frac{\sqrt{22}i}{44} w=\frac{\sqrt{22}i}{44}$$