Solve 3(3/2-x^2)={left(11/2-2x^2right)}^2

$$3( \frac{ 3 }{ 2 } - { x }^{ 2 } )= { \left( \frac{ 11 }{ 2 } -2 { x }^{ 2 } \right) }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{\sqrt[4]{103}e^{\frac{-\arctan(\frac{\sqrt{51}}{19})i+2\pi i}{2}}}{2}\approx -1.567198598+0.284800712i$$

$$x=\frac{\sqrt[4]{103}e^{-\frac{\arctan(\frac{\sqrt{51}}{19})i}{2}}}{2}\approx 1.567198598-0.284800712i$$

$$x=\frac{\sqrt[4]{103}e^{\frac{\arctan(\frac{\sqrt{51}}{19})i+2\pi i}{2}}}{2}\approx -1.567198598-0.284800712i$$

$$x=\frac{\sqrt[4]{103}e^{\frac{\arctan(\frac{\sqrt{51}}{19})i}{2}}}{2}\approx 1.567198598+0.284800712i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3 by \frac{3}{2}-x^{2}.

$$\frac{9}{2}-3x^{2}=\left(\frac{11}{2}-2x^{2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{11}{2}-2x^{2}\right)^{2}.

$$\frac{9}{2}-3x^{2}=\frac{121}{4}-22x^{2}+4\left(x^{2}\right)^{2}$$

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

$$\frac{9}{2}-3x^{2}=\frac{121}{4}-22x^{2}+4x^{4}$$

Subtract \frac{121}{4} from both sides.

$$\frac{9}{2}-3x^{2}-\frac{121}{4}=-22x^{2}+4x^{4}$$

Subtract \frac{121}{4} from \frac{9}{2} to get -\frac{103}{4}.

$$-\frac{103}{4}-3x^{2}=-22x^{2}+4x^{4}$$

Add 22x^{2} to both sides.

$$-\frac{103}{4}-3x^{2}+22x^{2}=4x^{4}$$

Combine -3x^{2} and 22x^{2} to get 19x^{2}.

$$-\frac{103}{4}+19x^{2}=4x^{4}$$

Subtract 4x^{4} from both sides.

$$-\frac{103}{4}+19x^{2}-4x^{4}=0$$

Substitute t for x^{2}.

$$-4t^{2}+19t-\frac{103}{4}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute -4 for a, 19 for b, and -\frac{103}{4} for c in the quadratic formula.

$$t=\frac{-19±\sqrt{19^{2}-4\left(-4\right)\left(-\frac{103}{4}\right)}}{-4\times 2}$$

Do the calculations.

$$t=\frac{-19±\sqrt{-51}}{-8}$$

Solve the equation t=\frac{-19±\sqrt{-51}}{-8} when ± is plus and when ± is minus.

$$t=\frac{-\sqrt{51}i+19}{8} t=\frac{19+\sqrt{51}i}{8}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=\frac{\sqrt[4]{103}e^{-\frac{\arctan(\frac{\sqrt{51}}{19})i}{2}}}{2} x=\frac{\sqrt[4]{103}e^{\frac{-\arctan(\frac{\sqrt{51}}{19})i+2\pi i}{2}}}{2} x=\frac{\sqrt[4]{103}e^{\frac{\arctan(\frac{\sqrt{51}}{19})i+2\pi i}{2}}}{2} x=\frac{\sqrt[4]{103}e^{\frac{\arctan(\frac{\sqrt{51}}{19})i}{2}}}{2}$$