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

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

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 \left(b-2i\right)\left(b+2i\right)b^{2}, the least common multiple of b^{2}+4,b^{2}.

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

Use the distributive property to multiply b^{2}+4 by \frac{25}{9}.

$$b^{2}\times 4+\frac{25}{9}b^{2}+\frac{100}{9}=\left(b-2i\right)\left(b+2i\right)b^{2}$$

Combine b^{2}\times 4 and \frac{25}{9}b^{2} to get \frac{61}{9}b^{2}.

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

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

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

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

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

Subtract b^{4} from both sides.

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

Subtract 4b^{2} from both sides.

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

Combine \frac{61}{9}b^{2} and -4b^{2} to get \frac{25}{9}b^{2}.

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

Substitute t for b^{2}.

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

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, \frac{25}{9} for b, and \frac{100}{9} for c in the quadratic formula.

$$t=\frac{-\frac{25}{9}±\sqrt{\left(\frac{25}{9}\right)^{2}-4\left(-1\right)\times \frac{100}{9}}}{-2}$$

Do the calculations.

$$t=\frac{-\frac{25}{9}±\frac{65}{9}}{-2}$$

Solve the equation t=\frac{-\frac{25}{9}±\frac{65}{9}}{-2} 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 b^{2}\left(b^{2}+4\right), the least common multiple of b^{2}+4,b^{2}.

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

Use the distributive property to multiply b^{2}+4 by \frac{25}{9}.

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

Combine b^{2}\times 4 and \frac{25}{9}b^{2} to get \frac{61}{9}b^{2}.

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

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

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

Subtract b^{4} from both sides.

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

Subtract 4b^{2} from both sides.

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

Combine \frac{61}{9}b^{2} and -4b^{2} to get \frac{25}{9}b^{2}.

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

Substitute t for b^{2}.

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

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, \frac{25}{9} for b, and \frac{100}{9} for c in the quadratic formula.

$$t=\frac{-\frac{25}{9}±\sqrt{\left(\frac{25}{9}\right)^{2}-4\left(-1\right)\times \frac{100}{9}}}{-2}$$

Do the calculations.

$$t=\frac{-\frac{25}{9}±\frac{65}{9}}{-2}$$

Solve the equation t=\frac{-\frac{25}{9}±\frac{65}{9}}{-2} 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.

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