Solve a/k(k-1)x^2+4x+1=0

$$\frac { a } { k } ( k - 1 ) x ^ { 2 } + 4 x + 1 = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{k\left(4x+1\right)}{\left(k-1\right)x^{2}}\text{, }&k\neq 0\text{ and }x\neq 0\text{ and }k\neq 1\\a\in \mathrm{C}\text{, }&k=1\text{ and }x=-\frac{1}{4}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by k.

$$a\left(k-1\right)x^{2}+4xk+k=0$$

Use the distributive property to multiply a by k-1.

$$\left(ak-a\right)x^{2}+4xk+k=0$$

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

$$akx^{2}-ax^{2}+4xk+k=0$$

Subtract 4xk from both sides. Anything subtracted from zero gives its negation.

$$akx^{2}-ax^{2}+k=-4xk$$

Subtract k from both sides.

$$akx^{2}-ax^{2}=-4xk-k$$

Combine all terms containing a.

$$\left(kx^{2}-x^{2}\right)a=-4xk-k$$

The equation is in standard form.

$$\left(kx^{2}-x^{2}\right)a=-4kx-k$$

Divide both sides by -x^{2}+kx^{2}.

$$\frac{\left(kx^{2}-x^{2}\right)a}{kx^{2}-x^{2}}=-\frac{k\left(4x+1\right)}{kx^{2}-x^{2}}$$

Dividing by -x^{2}+kx^{2} undoes the multiplication by -x^{2}+kx^{2}.

$$a=-\frac{k\left(4x+1\right)}{kx^{2}-x^{2}}$$

Divide -k\left(1+4x\right) by -x^{2}+kx^{2}.

$$a=-\frac{k\left(4x+1\right)}{\left(k-1\right)x^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{k\left(4x+1\right)}{\left(k-1\right)x^{2}}\text{, }&k\neq 0\text{ and }x\neq 0\text{ and }k\neq 1\\a\in \mathrm{R}\text{, }&k=1\text{ and }x=-\frac{1}{4}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by k.

$$a\left(k-1\right)x^{2}+4xk+k=0$$

Use the distributive property to multiply a by k-1.

$$\left(ak-a\right)x^{2}+4xk+k=0$$

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

$$akx^{2}-ax^{2}+4xk+k=0$$

Subtract 4xk from both sides. Anything subtracted from zero gives its negation.

$$akx^{2}-ax^{2}+k=-4xk$$

Subtract k from both sides.

$$akx^{2}-ax^{2}=-4xk-k$$

Combine all terms containing a.

$$\left(kx^{2}-x^{2}\right)a=-4xk-k$$

The equation is in standard form.

$$\left(kx^{2}-x^{2}\right)a=-4kx-k$$

Divide both sides by kx^{2}-x^{2}.

$$\frac{\left(kx^{2}-x^{2}\right)a}{kx^{2}-x^{2}}=-\frac{k\left(4x+1\right)}{kx^{2}-x^{2}}$$

Dividing by kx^{2}-x^{2} undoes the multiplication by kx^{2}-x^{2}.

$$a=-\frac{k\left(4x+1\right)}{kx^{2}-x^{2}}$$

Divide -k\left(1+4x\right) by kx^{2}-x^{2}.

$$a=-\frac{k\left(4x+1\right)}{\left(k-1\right)x^{2}}$$

Solve for k

$$\left\{\begin{matrix}k=\frac{ax^{2}}{ax^{2}+4x+1}\text{, }&x\neq 0\text{ and }a\neq 0\text{ and }a\neq -\frac{4x+1}{x^{2}}\\k\neq 0\text{, }&x=-\frac{1}{4}\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by k.

$$a\left(k-1\right)x^{2}+4xk+k=0$$

Use the distributive property to multiply a by k-1.

$$\left(ak-a\right)x^{2}+4xk+k=0$$

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

$$akx^{2}-ax^{2}+4xk+k=0$$

Add ax^{2} to both sides. Anything plus zero gives itself.

$$akx^{2}+4xk+k=ax^{2}$$

Combine all terms containing k.

$$\left(ax^{2}+4x+1\right)k=ax^{2}$$

Divide both sides by 4x+1+ax^{2}.

$$\frac{\left(ax^{2}+4x+1\right)k}{ax^{2}+4x+1}=\frac{ax^{2}}{ax^{2}+4x+1}$$

Dividing by 4x+1+ax^{2} undoes the multiplication by 4x+1+ax^{2}.

$$k=\frac{ax^{2}}{ax^{2}+4x+1}$$

Variable k cannot be equal to 0.

$$k=\frac{ax^{2}}{ax^{2}+4x+1}\text{, }k\neq 0$$