Solve (a-2)^2/-4a=a+1/2ap+1

$$\frac { ( a - 2 ) ^ { 2 } } { - 4 a } = \frac { a + 1 } { 2 a } p + 1$$

Solve for p

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

$$a\neq -1\text{ and }a\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4a, the least common multiple of -4a,2a.

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

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-2\right)^{2}.

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

To find the opposite of a^{2}-4a+4, find the opposite of each term.

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

Use the distributive property to multiply 2 by a+1.

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

Use the distributive property to multiply 2a+2 by p.

$$-a^{2}+4a-4=2ap+2p+4a$$

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

$$2ap+2p+4a=-a^{2}+4a-4$$

Subtract 4a from both sides.

$$2ap+2p=-a^{2}+4a-4-4a$$

Combine 4a and -4a to get 0.

$$2ap+2p=-a^{2}-4$$

Combine all terms containing p.

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

Divide both sides by 2+2a.

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

Dividing by 2+2a undoes the multiplication by 2+2a.

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

Divide -a^{2}-4 by 2+2a.

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

Solve for a

$$\left\{\begin{matrix}a=\sqrt{p^{2}-2p-4}-p\text{, }&p\geq \sqrt{5}+1\text{ or }p\leq 1-\sqrt{5}\\a=-\sqrt{p^{2}-2p-4}-p\text{, }&p\geq \sqrt{5}+1\text{ or }\left(p\neq -2\text{ and }p\leq 1-\sqrt{5}\right)\end{matrix}\right.$$