Solve 1/2(x-1)^2-1=1/2m-1

$$\frac { 1 } { 2 } ( x - 1 ) ^ { 2 } - 1 = \frac { 1 } { 2 } m - 1$$

Solve for m

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

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Use the distributive property to multiply \frac{1}{2} by x^{2}-2x+1.

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

Subtract 1 from \frac{1}{2} to get -\frac{1}{2}.

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

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

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

Add 1 to both sides.

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

Add -\frac{1}{2} and 1 to get \frac{1}{2}.

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

The equation is in standard form.

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

Multiply both sides by 2.

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

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

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

Divide \frac{\left(-1+x\right)^{2}}{2} by \frac{1}{2} by multiplying \frac{\left(-1+x\right)^{2}}{2} by the reciprocal of \frac{1}{2}.

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

Solve for x (complex solution)

$$x=-\sqrt{m}+1$$

$$x=\sqrt{m}+1$$

Solve for x

$$x=-\sqrt{m}+1$$

$$x=\sqrt{m}+1\text{, }m\geq 0$$