Solve 4/b^2+4+25/9b^2=1

$$\frac { 4 } { b ^ { 2 } + 4 } + \frac { 25 } { 9 b ^ { 2 } } = 1$$

Solve for b (complex solution)

$$b=\frac{2\sqrt{5}i}{3}\approx 1.490711985i$$

$$b=-\frac{2\sqrt{5}i}{3}\approx -0-1.490711985i$$

$$b=-\sqrt{5}\approx -2.236067977$$

$$b=\sqrt{5}\approx 2.236067977$$

Steps Using the Quadratic Formula

Variable b cannot be equal to any of the values -2i,0,2i since division by zero is not defined. Multiply both sides of the equation by 9\left(b-2i\right)\left(b+2i\right)b^{2}, the least common multiple of b^{2}+4,9b^{2}.

$$9b^{2}\times 4+\left(b^{2}+4\right)\times 25=9\left(b-2i\right)\left(b+2i\right)b^{2}$$

Multiply 9 and 4 to get 36.

$$36b^{2}+\left(b^{2}+4\right)\times 25=9\left(b-2i\right)\left(b+2i\right)b^{2}$$

Use the distributive property to multiply b^{2}+4 by 25.

$$36b^{2}+25b^{2}+100=9\left(b-2i\right)\left(b+2i\right)b^{2}$$

Combine 36b^{2} and 25b^{2} to get 61b^{2}.

$$61b^{2}+100=9\left(b-2i\right)\left(b+2i\right)b^{2}$$

Use the distributive property to multiply 9 by b-2i.

$$61b^{2}+100=\left(9b-18i\right)\left(b+2i\right)b^{2}$$

Use the distributive property to multiply 9b-18i by b+2i and combine like terms.

$$61b^{2}+100=\left(9b^{2}+36\right)b^{2}$$

Use the distributive property to multiply 9b^{2}+36 by b^{2}.

$$61b^{2}+100=9b^{4}+36b^{2}$$

Subtract 9b^{4} from both sides.

$$61b^{2}+100-9b^{4}=36b^{2}$$

Subtract 36b^{2} from both sides.

$$61b^{2}+100-9b^{4}-36b^{2}=0$$

Combine 61b^{2} and -36b^{2} to get 25b^{2}.

$$25b^{2}+100-9b^{4}=0$$

Substitute t for b^{2}.

$$-9t^{2}+25t+100=0$$

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 -9 for a, 25 for b, and 100 for c in the quadratic formula.

$$t=\frac{-25±\sqrt{25^{2}-4\left(-9\right)\times 100}}{-9\times 2}$$

Do the calculations.

$$t=\frac{-25±65}{-18}$$

Solve the equation t=\frac{-25±65}{-18} when ± is plus and when ± is minus.

$$t=-\frac{20}{9} t=5$$

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

$$b=-\frac{2\sqrt{5}i}{3} b=\frac{2\sqrt{5}i}{3} b=-\sqrt{5} b=\sqrt{5}$$

Solve for b

$$b=\sqrt{5}\approx 2.236067977$$

$$b=-\sqrt{5}\approx -2.236067977$$

Steps Using the Quadratic Formula

Variable b cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 9b^{2}\left(b^{2}+4\right), the least common multiple of b^{2}+4,9b^{2}.

$$9b^{2}\times 4+\left(b^{2}+4\right)\times 25=9b^{2}\left(b^{2}+4\right)$$

Multiply 9 and 4 to get 36.

$$36b^{2}+\left(b^{2}+4\right)\times 25=9b^{2}\left(b^{2}+4\right)$$

Use the distributive property to multiply b^{2}+4 by 25.

$$36b^{2}+25b^{2}+100=9b^{2}\left(b^{2}+4\right)$$

Combine 36b^{2} and 25b^{2} to get 61b^{2}.

$$61b^{2}+100=9b^{2}\left(b^{2}+4\right)$$

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

$$61b^{2}+100=9b^{4}+36b^{2}$$

Subtract 9b^{4} from both sides.

$$61b^{2}+100-9b^{4}=36b^{2}$$

Subtract 36b^{2} from both sides.

$$61b^{2}+100-9b^{4}-36b^{2}=0$$

Combine 61b^{2} and -36b^{2} to get 25b^{2}.

$$25b^{2}+100-9b^{4}=0$$

Substitute t for b^{2}.

$$-9t^{2}+25t+100=0$$

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 -9 for a, 25 for b, and 100 for c in the quadratic formula.

$$t=\frac{-25±\sqrt{25^{2}-4\left(-9\right)\times 100}}{-9\times 2}$$

Do the calculations.

$$t=\frac{-25±65}{-18}$$

Solve the equation t=\frac{-25±65}{-18} when ± is plus and when ± is minus.

$$t=-\frac{20}{9} t=5$$

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

$$b=\sqrt{5} b=-\sqrt{5}$$