Solve (t-3)x^2-5x-6=0

$$( t - 3 ) x ^ { 2 } - 5 x - 6 = 0$$

Solve for t

$$t=3+\frac{5}{x}+\frac{6}{x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply t-3 by x^{2}.

$$tx^{2}-3x^{2}-5x-6=0$$

Add 3x^{2} to both sides. Anything plus zero gives itself.

$$tx^{2}-5x-6=3x^{2}$$

Add 5x to both sides.

$$tx^{2}-6=3x^{2}+5x$$

Add 6 to both sides.

$$tx^{2}=3x^{2}+5x+6$$

The equation is in standard form.

$$x^{2}t=3x^{2}+5x+6$$

Divide both sides by x^{2}.

$$\frac{x^{2}t}{x^{2}}=\frac{3x^{2}+5x+6}{x^{2}}$$

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

$$t=\frac{3x^{2}+5x+6}{x^{2}}$$

Divide 3x^{2}+5x+6 by x^{2}.

$$t=3+\frac{5x+6}{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{24t-47}+5}{2\left(t-3\right)}\text{; }x=\frac{-\sqrt{24t-47}+5}{2\left(t-3\right)}\text{, }&t\neq 3\\x=-\frac{6}{5}\text{, }&t=3\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{24t-47}+5}{2\left(t-3\right)}\text{; }x=\frac{-\sqrt{24t-47}+5}{2\left(t-3\right)}\text{, }&t\neq 3\text{ and }t\geq \frac{47}{24}\\x=-\frac{6}{5}\text{, }&t=3\end{matrix}\right.$$