Solve 25/9-x^2-16/x^2=1

$$\frac { 25 } { 9 - x ^ { 2 } } - \frac { 16 } { x ^ { 2 } } = 1$$

Solve for x (complex solution)

$$x=6i$$

$$x=-6i$$

$$x=-2$$

$$x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right)x^{2}, the least common multiple of 9-x^{2},x^{2}.

$$-x^{2}\times 25-\left(x^{2}-9\right)\times 16=\left(x-3\right)\left(x+3\right)x^{2}$$

Multiply -1 and 25 to get -25.

$$-25x^{2}-\left(x^{2}-9\right)\times 16=\left(x-3\right)\left(x+3\right)x^{2}$$

Use the distributive property to multiply x^{2}-9 by 16.

$$-25x^{2}-\left(16x^{2}-144\right)=\left(x-3\right)\left(x+3\right)x^{2}$$

To find the opposite of 16x^{2}-144, find the opposite of each term.

$$-25x^{2}-16x^{2}+144=\left(x-3\right)\left(x+3\right)x^{2}$$

Combine -25x^{2} and -16x^{2} to get -41x^{2}.

$$-41x^{2}+144=\left(x-3\right)\left(x+3\right)x^{2}$$

Use the distributive property to multiply x-3 by x+3 and combine like terms.

$$-41x^{2}+144=\left(x^{2}-9\right)x^{2}$$

Use the distributive property to multiply x^{2}-9 by x^{2}.

$$-41x^{2}+144=x^{4}-9x^{2}$$

Subtract x^{4} from both sides.

$$-41x^{2}+144-x^{4}=-9x^{2}$$

Add 9x^{2} to both sides.

$$-41x^{2}+144-x^{4}+9x^{2}=0$$

Combine -41x^{2} and 9x^{2} to get -32x^{2}.

$$-32x^{2}+144-x^{4}=0$$

Substitute t for x^{2}.

$$-t^{2}-32t+144=0$$

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, -32 for b, and 144 for c in the quadratic formula.

$$t=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-1\right)\times 144}}{-2}$$

Do the calculations.

$$t=\frac{32±40}{-2}$$

Solve the equation t=\frac{32±40}{-2} when ± is plus and when ± is minus.

$$t=-36 t=4$$

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

$$x=-6i x=6i x=-2 x=2$$

Solve for x

$$x=2$$

$$x=-2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right)x^{2}, the least common multiple of 9-x^{2},x^{2}.

$$-x^{2}\times 25-\left(x^{2}-9\right)\times 16=\left(x-3\right)\left(x+3\right)x^{2}$$

Multiply -1 and 25 to get -25.

$$-25x^{2}-\left(x^{2}-9\right)\times 16=\left(x-3\right)\left(x+3\right)x^{2}$$

Use the distributive property to multiply x^{2}-9 by 16.

$$-25x^{2}-\left(16x^{2}-144\right)=\left(x-3\right)\left(x+3\right)x^{2}$$

To find the opposite of 16x^{2}-144, find the opposite of each term.

$$-25x^{2}-16x^{2}+144=\left(x-3\right)\left(x+3\right)x^{2}$$

Combine -25x^{2} and -16x^{2} to get -41x^{2}.

$$-41x^{2}+144=\left(x-3\right)\left(x+3\right)x^{2}$$

Use the distributive property to multiply x-3 by x+3 and combine like terms.

$$-41x^{2}+144=\left(x^{2}-9\right)x^{2}$$

Use the distributive property to multiply x^{2}-9 by x^{2}.

$$-41x^{2}+144=x^{4}-9x^{2}$$

Subtract x^{4} from both sides.

$$-41x^{2}+144-x^{4}=-9x^{2}$$

Add 9x^{2} to both sides.

$$-41x^{2}+144-x^{4}+9x^{2}=0$$

Combine -41x^{2} and 9x^{2} to get -32x^{2}.

$$-32x^{2}+144-x^{4}=0$$

Substitute t for x^{2}.

$$-t^{2}-32t+144=0$$

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, -32 for b, and 144 for c in the quadratic formula.

$$t=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-1\right)\times 144}}{-2}$$

Do the calculations.

$$t=\frac{32±40}{-2}$$

Solve the equation t=\frac{32±40}{-2} when ± is plus and when ± is minus.

$$t=-36 t=4$$

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

$$x=2 x=-2$$