Solve x^2-left(1-pright)x-1=3x-p

$${ x }^{ 2 } - \left( 1-p \right) x-1 = 3x-p$$

Solve for p

$$p=\frac{1+4x-x^{2}}{x+1}$$

$$x\neq -1$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1-p by x.

$$x^{2}-\left(x-px\right)-1=3x-p$$

To find the opposite of x-px, find the opposite of each term.

$$x^{2}-x+px-1=3x-p$$

Add p to both sides.

$$x^{2}-x+px-1+p=3x$$

Subtract x^{2} from both sides.

$$-x+px-1+p=3x-x^{2}$$

Add x to both sides.

$$px-1+p=3x-x^{2}+x$$

Combine 3x and x to get 4x.

$$px-1+p=4x-x^{2}$$

Add 1 to both sides.

$$px+p=4x-x^{2}+1$$

Combine all terms containing p.

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

The equation is in standard form.

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

Divide both sides by x+1.

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

Dividing by x+1 undoes the multiplication by x+1.

$$p=\frac{1+4x-x^{2}}{x+1}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{\left(p-10\right)\left(p-2\right)}-p+4}{2}$$

$$x=\frac{-\sqrt{\left(p-10\right)\left(p-2\right)}-p+4}{2}$$

Solve for x

$$x=\frac{\sqrt{\left(p-10\right)\left(p-2\right)}-p+4}{2}$$

$$x=\frac{-\sqrt{\left(p-10\right)\left(p-2\right)}-p+4}{2}\text{, }p\leq 2\text{ or }p\geq 10$$