Solve (4+3/k^2)y^2+b/ky-9=0

$$( 4 + \frac { 3 } { k ^ { 2 } } ) y ^ { 2 } + \frac { b } { k } y - 9 = 0$$

Solve for b

$$b=\frac{9k^{2}-4\left(ky\right)^{2}-3y^{2}}{ky}$$

$$k\neq 0\text{ and }y\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by k^{2}, the least common multiple of k^{2},k.

$$\left(4+\frac{3}{k^{2}}\right)y^{2}k^{2}+kby+k^{2}\left(-9\right)=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{k^{2}}{k^{2}}.

$$\left(\frac{4k^{2}}{k^{2}}+\frac{3}{k^{2}}\right)y^{2}k^{2}+kby+k^{2}\left(-9\right)=0$$

Since \frac{4k^{2}}{k^{2}} and \frac{3}{k^{2}} have the same denominator, add them by adding their numerators.

$$\frac{4k^{2}+3}{k^{2}}y^{2}k^{2}+kby+k^{2}\left(-9\right)=0$$

Express \frac{4k^{2}+3}{k^{2}}y^{2} as a single fraction.

$$\frac{\left(4k^{2}+3\right)y^{2}}{k^{2}}k^{2}+kby+k^{2}\left(-9\right)=0$$

Express \frac{\left(4k^{2}+3\right)y^{2}}{k^{2}}k^{2} as a single fraction.

$$\frac{\left(4k^{2}+3\right)y^{2}k^{2}}{k^{2}}+kby+k^{2}\left(-9\right)=0$$

Cancel out k^{2} in both numerator and denominator.

$$y^{2}\left(4k^{2}+3\right)+kby+k^{2}\left(-9\right)=0$$

Use the distributive property to multiply y^{2} by 4k^{2}+3.

$$4y^{2}k^{2}+3y^{2}+kby+k^{2}\left(-9\right)=0$$

Subtract 4y^{2}k^{2} from both sides. Anything subtracted from zero gives its negation.

$$3y^{2}+kby+k^{2}\left(-9\right)=-4y^{2}k^{2}$$

Subtract 3y^{2} from both sides.

$$kby+k^{2}\left(-9\right)=-4y^{2}k^{2}-3y^{2}$$

Subtract k^{2}\left(-9\right) from both sides.

$$kby=-4y^{2}k^{2}-3y^{2}-k^{2}\left(-9\right)$$

Multiply -1 and -9 to get 9.

$$kby=-4y^{2}k^{2}-3y^{2}+9k^{2}$$

The equation is in standard form.

$$kyb=9k^{2}-3y^{2}-4k^{2}y^{2}$$

Divide both sides by ky.

$$\frac{kyb}{ky}=\frac{9k^{2}-3y^{2}-4k^{2}y^{2}}{ky}$$

Dividing by ky undoes the multiplication by ky.

$$b=\frac{9k^{2}-3y^{2}-4k^{2}y^{2}}{ky}$$

Divide -4y^{2}k^{2}-3y^{2}+9k^{2} by ky.

$$b=-4ky+\frac{9k}{y}-\frac{3y}{k}$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=\frac{\sqrt{y^{2}\left(108+b^{2}-48y^{2}\right)}-by}{2\left(4y^{2}-9\right)}\text{; }k=-\frac{\sqrt{y^{2}\left(108+b^{2}-48y^{2}\right)}+by}{2\left(4y^{2}-9\right)}\text{, }&y\neq \frac{3}{2}\text{ and }y\neq -\frac{3}{2}\text{ and }y\neq 0\\k=-\frac{3y}{b}\text{, }&b\neq 0\text{ and }\left(y=-\frac{3}{2}\text{ or }y=\frac{3}{2}\right)\end{matrix}\right.$$

Solve for k

$$\left\{\begin{matrix}k=\frac{y\left(\sqrt{108+b^{2}-48y^{2}}-b\right)}{2\left(4y^{2}-9\right)}\text{; }k=-\frac{y\left(\sqrt{108+b^{2}-48y^{2}}+b\right)}{2\left(4y^{2}-9\right)}\text{, }&y\neq 0\text{ and }|y|\neq \frac{3}{2}\text{ and }|y|\leq \frac{\sqrt{3b^{2}+324}}{12}\\k=-\frac{3y}{b}\text{, }&b\neq 0\text{ and }|y|=\frac{3}{2}\end{matrix}\right.$$