Solve 4:f(x)=3x^2-18x+25/x^2-6x+8

$$4 : f ( x ) = \frac { 3 x ^ { 2 } - 18 x + 25 } { x ^ { 2 } - 6 x + 8 }$$

Solve for f

$$f=-\frac{4x\left(x^{2}-6x+8\right)}{-3x^{2}+18x-25}$$

$$x\neq 2\text{ and }x\neq 4\text{ and }x\neq 0\text{ and }x\neq \frac{\sqrt{6}}{3}+3\text{ and }x\neq -\frac{\sqrt{6}}{3}+3$$

Steps for Solving Linear Equation

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f\left(x-4\right)\left(x-2\right), the least common multiple of f,x^{2}-6x+8.

$$\left(x-4\right)\left(x-2\right)\times 4x=f\left(3x^{2}-18x+25\right)$$

Use the distributive property to multiply x-4 by x-2 and combine like terms.

$$\left(x^{2}-6x+8\right)\times 4x=f\left(3x^{2}-18x+25\right)$$

Use the distributive property to multiply x^{2}-6x+8 by 4.

$$\left(4x^{2}-24x+32\right)x=f\left(3x^{2}-18x+25\right)$$

Use the distributive property to multiply 4x^{2}-24x+32 by x.

$$4x^{3}-24x^{2}+32x=f\left(3x^{2}-18x+25\right)$$

Use the distributive property to multiply f by 3x^{2}-18x+25.

$$4x^{3}-24x^{2}+32x=3fx^{2}-18fx+25f$$

Swap sides so that all variable terms are on the left hand side.

$$3fx^{2}-18fx+25f=4x^{3}-24x^{2}+32x$$

Combine all terms containing f.

$$\left(3x^{2}-18x+25\right)f=4x^{3}-24x^{2}+32x$$

Divide both sides by 3x^{2}-18x+25.

$$\frac{\left(3x^{2}-18x+25\right)f}{3x^{2}-18x+25}=\frac{4x\left(x-4\right)\left(x-2\right)}{3x^{2}-18x+25}$$

Dividing by 3x^{2}-18x+25 undoes the multiplication by 3x^{2}-18x+25.

$$f=\frac{4x\left(x-4\right)\left(x-2\right)}{3x^{2}-18x+25}$$

Variable f cannot be equal to 0.

$$f=\frac{4x\left(x-4\right)\left(x-2\right)}{3x^{2}-18x+25}\text{, }f\neq 0$$