Solve 4/4-b^2-9/b^2=1

$$\frac{ 4 }{ 4- { b }^{ 2 } } - \frac{ 9 }{ { b }^{ 2 } } = 1$$

Solve for b (complex solution)

$$b=2\sqrt{3}i\approx 3.464101615i$$

$$b=-2\sqrt{3}i\approx -0-3.464101615i$$

$$b=-\sqrt{3}\approx -1.732050808$$

$$b=\sqrt{3}\approx 1.732050808$$

Steps Using the Quadratic Formula

Variable b cannot be equal to any of the values -2,0,2 since division by zero is not defined. Multiply both sides of the equation by \left(b-2\right)\left(b+2\right)b^{2}, the least common multiple of 4-b^{2},b^{2}.

$$-b^{2}\times 4-\left(b^{2}-4\right)\times 9=\left(b-2\right)\left(b+2\right)b^{2}$$

Multiply -1 and 4 to get -4.

$$-4b^{2}-\left(b^{2}-4\right)\times 9=\left(b-2\right)\left(b+2\right)b^{2}$$

Use the distributive property to multiply b^{2}-4 by 9.

$$-4b^{2}-\left(9b^{2}-36\right)=\left(b-2\right)\left(b+2\right)b^{2}$$

To find the opposite of 9b^{2}-36, find the opposite of each term.

$$-4b^{2}-9b^{2}+36=\left(b-2\right)\left(b+2\right)b^{2}$$

Combine -4b^{2} and -9b^{2} to get -13b^{2}.

$$-13b^{2}+36=\left(b-2\right)\left(b+2\right)b^{2}$$

Use the distributive property to multiply b-2 by b+2 and combine like terms.

$$-13b^{2}+36=\left(b^{2}-4\right)b^{2}$$

Use the distributive property to multiply b^{2}-4 by b^{2}.

$$-13b^{2}+36=b^{4}-4b^{2}$$

Subtract b^{4} from both sides.

$$-13b^{2}+36-b^{4}=-4b^{2}$$

Add 4b^{2} to both sides.

$$-13b^{2}+36-b^{4}+4b^{2}=0$$

Combine -13b^{2} and 4b^{2} to get -9b^{2}.

$$-9b^{2}+36-b^{4}=0$$

Substitute t for b^{2}.

$$-t^{2}-9t+36=0$$

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, -9 for b, and 36 for c in the quadratic formula.

$$t=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-1\right)\times 36}}{-2}$$

Do the calculations.

$$t=\frac{9±15}{-2}$$

Solve the equation t=\frac{9±15}{-2} when ± is plus and when ± is minus.

$$t=-12 t=3$$

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

$$b=-2\sqrt{3}i b=2\sqrt{3}i b=-\sqrt{3} b=\sqrt{3}$$

Solve for b

$$b=\sqrt{3}\approx 1.732050808$$

$$b=-\sqrt{3}\approx -1.732050808$$

Steps Using the Quadratic Formula

Variable b cannot be equal to any of the values -2,0,2 since division by zero is not defined. Multiply both sides of the equation by \left(b-2\right)\left(b+2\right)b^{2}, the least common multiple of 4-b^{2},b^{2}.

$$-b^{2}\times 4-\left(b^{2}-4\right)\times 9=\left(b-2\right)\left(b+2\right)b^{2}$$

Multiply -1 and 4 to get -4.

$$-4b^{2}-\left(b^{2}-4\right)\times 9=\left(b-2\right)\left(b+2\right)b^{2}$$

Use the distributive property to multiply b^{2}-4 by 9.

$$-4b^{2}-\left(9b^{2}-36\right)=\left(b-2\right)\left(b+2\right)b^{2}$$

To find the opposite of 9b^{2}-36, find the opposite of each term.

$$-4b^{2}-9b^{2}+36=\left(b-2\right)\left(b+2\right)b^{2}$$

Combine -4b^{2} and -9b^{2} to get -13b^{2}.

$$-13b^{2}+36=\left(b-2\right)\left(b+2\right)b^{2}$$

Use the distributive property to multiply b-2 by b+2 and combine like terms.

$$-13b^{2}+36=\left(b^{2}-4\right)b^{2}$$

Use the distributive property to multiply b^{2}-4 by b^{2}.

$$-13b^{2}+36=b^{4}-4b^{2}$$

Subtract b^{4} from both sides.

$$-13b^{2}+36-b^{4}=-4b^{2}$$

Add 4b^{2} to both sides.

$$-13b^{2}+36-b^{4}+4b^{2}=0$$

Combine -13b^{2} and 4b^{2} to get -9b^{2}.

$$-9b^{2}+36-b^{4}=0$$

Substitute t for b^{2}.

$$-t^{2}-9t+36=0$$

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, -9 for b, and 36 for c in the quadratic formula.

$$t=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-1\right)\times 36}}{-2}$$

Do the calculations.

$$t=\frac{9±15}{-2}$$

Solve the equation t=\frac{9±15}{-2} when ± is plus and when ± is minus.

$$t=-12 t=3$$

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

$$b=\sqrt{3} b=-\sqrt{3}$$