Solve y=(x^2+5/2)^2-(7/2)^2

$$y = ( x ^ { 2 } + \frac { 5 } { 2 } ) ^ { 2 } - ( \frac { 7 } { 2 } ) ^ { 2 }$$

Solve for y

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

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+\frac{5}{2}\right)^{2}.

$$y=\left(x^{2}\right)^{2}+5x^{2}+\frac{25}{4}-\left(\frac{7}{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$y=x^{4}+5x^{2}+\frac{25}{4}-\left(\frac{7}{2}\right)^{2}$$

Calculate \frac{7}{2} to the power of 2 and get \frac{49}{4}.

$$y=x^{4}+5x^{2}+\frac{25}{4}-\frac{49}{4}$$

Subtract \frac{49}{4} from \frac{25}{4} to get -6.

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

Solve for x (complex solution)

$$x=\frac{\sqrt{2\sqrt{4y+49}-10}}{2}$$

$$x=-\frac{\sqrt{2\sqrt{4y+49}-10}}{2}$$

$$x=-\frac{\sqrt{-2\sqrt{4y+49}-10}}{2}$$

$$x=\frac{\sqrt{-2\sqrt{4y+49}-10}}{2}$$

Solve for x

$$x=-\frac{\sqrt{2\left(\sqrt{4y+49}-5\right)}}{2}$$

$$x=\frac{\sqrt{2\left(\sqrt{4y+49}-5\right)}}{2}\text{, }y\geq -6$$