Solve f(x)=-x^2+2ax+1=a

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

Solve for a

$$a=-\frac{1-x^{2}}{2x-1}$$

$$x\neq \frac{1}{2}$$

Steps for Solving Linear Equation

Subtract a from both sides.

$$-x^{2}+2ax+1-a=0$$

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

$$2ax+1-a=x^{2}$$

Subtract 1 from both sides.

$$2ax-a=x^{2}-1$$

Combine all terms containing a.

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

Divide both sides by 2x-1.

$$\frac{\left(2x-1\right)a}{2x-1}=\frac{x^{2}-1}{2x-1}$$

Dividing by 2x-1 undoes the multiplication by 2x-1.

$$a=\frac{x^{2}-1}{2x-1}$$

Solve for x

$$x=-\sqrt{a^{2}-a+1}+a$$

$$x=\sqrt{a^{2}-a+1}+a$$