Solve y=1/2x^2+1/3x^2-1/2x^2+1/4

$$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 2 } - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 4 }$$

Solve for y

$$y=\frac{x^{2}}{3}+\frac{1}{4}$$

Solution Steps

Combine \frac{1}{2}x^{2} and \frac{1}{3}x^{2} to get \frac{5}{6}x^{2}.

$$y=\frac{5}{6}x^{2}-\frac{1}{2}x^{2}+\frac{1}{4}$$

Combine \frac{5}{6}x^{2} and -\frac{1}{2}x^{2} to get \frac{1}{3}x^{2}.

$$y=\frac{1}{3}x^{2}+\frac{1}{4}$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{12y-3}}{2}$$

$$x=\frac{\sqrt{12y-3}}{2}$$

Solve for x

$$x=\frac{\sqrt{12y-3}}{2}$$

$$x=-\frac{\sqrt{12y-3}}{2}\text{, }y\geq \frac{1}{4}$$