Solve (1/2x^2-6)-2x=-x+t

$$( \frac { 1 } { 2 } x ^ { 2 } - 6 ) - 2 x = - x + t$$

Solve for t

$$t=\frac{x^{2}-2x-12}{2}$$

Solution Steps

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

$$-x+t=\frac{1}{2}x^{2}-6-2x$$

Add x to both sides.

$$t=\frac{1}{2}x^{2}-6-2x+x$$

Combine -2x and x to get -x.

$$t=\frac{1}{2}x^{2}-6-x$$

Solve for x (complex solution)

$$x=\sqrt{2t+13}+1$$

$$x=-\sqrt{2t+13}+1$$

Solve for x

$$x=\sqrt{2t+13}+1$$

$$x=-\sqrt{2t+13}+1\text{, }t\geq -\frac{13}{2}$$