Solve (x-8)(x+4)=(x-a)^2+1

$$( x - 8 ) ( x + 4 ) = ( x - a ) ^ { 2 } + 1$$

Solve for x

$$x=\frac{a^{2}+33}{2\left(a-2\right)}$$

$$a\neq 2$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-8 by x+4 and combine like terms.

$$x^{2}-4x-32=\left(x-a\right)^{2}+1$$

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

$$x^{2}-4x-32=x^{2}-2xa+a^{2}+1$$

Subtract x^{2} from both sides.

$$x^{2}-4x-32-x^{2}=-2xa+a^{2}+1$$

Combine x^{2} and -x^{2} to get 0.

$$-4x-32=-2xa+a^{2}+1$$

Add 2xa to both sides.

$$-4x-32+2xa=a^{2}+1$$

Add 32 to both sides.

$$-4x+2xa=a^{2}+1+32$$

Add 1 and 32 to get 33.

$$-4x+2xa=a^{2}+33$$

Combine all terms containing x.

$$\left(-4+2a\right)x=a^{2}+33$$

The equation is in standard form.

$$\left(2a-4\right)x=a^{2}+33$$

Divide both sides by -4+2a.

$$\frac{\left(2a-4\right)x}{2a-4}=\frac{a^{2}+33}{2a-4}$$

Dividing by -4+2a undoes the multiplication by -4+2a.

$$x=\frac{a^{2}+33}{2a-4}$$

Divide a^{2}+33 by -4+2a.

$$x=\frac{a^{2}+33}{2\left(a-2\right)}$$

Solve for a (complex solution)

$$a=\sqrt{x^{2}-4x-33}+x$$

$$a=-\sqrt{x^{2}-4x-33}+x$$

Solve for a

$$a=\sqrt{x^{2}-4x-33}+x$$

$$a=-\sqrt{x^{2}-4x-33}+x\text{, }x\geq \sqrt{37}+2\text{ or }x\leq 2-\sqrt{37}$$