Solve 2x^2-1=1/2x^2+1/2

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

Solve for x

$$x=1$$

$$x=-1$$

Steps Using Difference of Squares

Subtract \frac{1}{2}x^{2} from both sides.

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

Combine 2x^{2} and -\frac{1}{2}x^{2} to get \frac{3}{2}x^{2}.

$$\frac{3}{2}x^{2}-1=\frac{1}{2}$$

Subtract \frac{1}{2} from both sides.

$$\frac{3}{2}x^{2}-1-\frac{1}{2}=0$$

Subtract \frac{1}{2} from -1 to get -\frac{3}{2}.

$$\frac{3}{2}x^{2}-\frac{3}{2}=0$$

Divide both sides by \frac{3}{2}.

$$x^{2}-1=0$$

Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-1\right)\left(x+1\right)=0$$

To find equation solutions, solve x-1=0 and x+1=0.

$$x=1 x=-1$$

Steps by Finding Square Root

Subtract \frac{1}{2}x^{2} from both sides.

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

Combine 2x^{2} and -\frac{1}{2}x^{2} to get \frac{3}{2}x^{2}.

$$\frac{3}{2}x^{2}-1=\frac{1}{2}$$

Add 1 to both sides.

$$\frac{3}{2}x^{2}=\frac{1}{2}+1$$

Add \frac{1}{2} and 1 to get \frac{3}{2}.

$$\frac{3}{2}x^{2}=\frac{3}{2}$$

Multiply both sides by \frac{2}{3}, the reciprocal of \frac{3}{2}.

$$x^{2}=\frac{3}{2}\times \frac{2}{3}$$

Multiply \frac{3}{2} and \frac{2}{3} to get 1.

$$x^{2}=1$$

Take the square root of both sides of the equation.

$$x=1 x=-1$$

Steps Using the Quadratic Formula

Subtract \frac{1}{2}x^{2} from both sides.

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

Combine 2x^{2} and -\frac{1}{2}x^{2} to get \frac{3}{2}x^{2}.

$$\frac{3}{2}x^{2}-1=\frac{1}{2}$$

Subtract \frac{1}{2} from both sides.

$$\frac{3}{2}x^{2}-1-\frac{1}{2}=0$$

Subtract \frac{1}{2} from -1 to get -\frac{3}{2}.

$$\frac{3}{2}x^{2}-\frac{3}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{2} for a, 0 for b, and -\frac{3}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{3}{2}\left(-\frac{3}{2}\right)}}{2\times \frac{3}{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{3}{2}\left(-\frac{3}{2}\right)}}{2\times \frac{3}{2}}$$

Multiply -4 times \frac{3}{2}.

$$x=\frac{0±\sqrt{-6\left(-\frac{3}{2}\right)}}{2\times \frac{3}{2}}$$

Multiply -6 times -\frac{3}{2}.

$$x=\frac{0±\sqrt{9}}{2\times \frac{3}{2}}$$

Take the square root of 9.

$$x=\frac{0±3}{2\times \frac{3}{2}}$$

Multiply 2 times \frac{3}{2}.

$$x=\frac{0±3}{3}$$

Now solve the equation x=\frac{0±3}{3} when ± is plus. Divide 3 by 3.

$$x=1$$

Now solve the equation x=\frac{0±3}{3} when ± is minus. Divide -3 by 3.

$$x=-1$$

The equation is now solved.

$$x=1 x=-1$$