Solve 4+3x^2/sqrt{7}=3+x^2/sqrt[3]{9}

$$\frac { 4 + 3 x ^ { 2 } } { \sqrt { 7 } } = \frac { 3 + x ^ { 2 } } { \sqrt[ 3 ] { 9 } }$$

Solve for x (complex solution)

$$x=-i\sqrt{\frac{4\times 3^{\frac{2}{3}}-3\sqrt{7}}{3\times 3^{\frac{2}{3}}-\sqrt{7}}}\approx -0-0.326457204i$$

$$x=i\sqrt{\frac{4\times 3^{\frac{2}{3}}-3\sqrt{7}}{3\times 3^{\frac{2}{3}}-\sqrt{7}}}\approx 0.326457204i$$

Steps by Finding Square Root

Rationalize the denominator of \frac{4+3x^{2}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\left(4+3x^{2}\right)\sqrt{7}}{\left(\sqrt{7}\right)^{2}}=\frac{3+x^{2}}{\sqrt[3]{9}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(4+3x^{2}\right)\sqrt{7}}{7}=\frac{3+x^{2}}{\sqrt[3]{9}}$$

Use the distributive property to multiply 4+3x^{2} by \sqrt{7}.

$$\frac{4\sqrt{7}+3x^{2}\sqrt{7}}{7}=\frac{3+x^{2}}{\sqrt[3]{9}}$$

Subtract \frac{3+x^{2}}{\sqrt[3]{9}} from both sides.

$$\frac{4\sqrt{7}+3x^{2}\sqrt{7}}{7}-\frac{3+x^{2}}{\sqrt[3]{9}}=0$$

Multiply both sides of the equation by 7.

$$4\sqrt{7}+3x^{2}\sqrt{7}-7\times 3^{-\frac{2}{3}}\left(3+x^{2}\right)=0$$

Reorder the terms.

$$-7\times 3^{-\frac{2}{3}}\left(x^{2}+3\right)+3\sqrt{7}x^{2}+4\sqrt{7}=0$$

Use the distributive property to multiply -7\times 3^{-\frac{2}{3}} by x^{2}+3.

$$-7\times 3^{-\frac{2}{3}}x^{2}-21\times 3^{-\frac{2}{3}}+3\sqrt{7}x^{2}+4\sqrt{7}=0$$

Add 21\times 3^{-\frac{2}{3}} to both sides. Anything plus zero gives itself.

$$-7\times 3^{-\frac{2}{3}}x^{2}+3\sqrt{7}x^{2}+4\sqrt{7}=21\times 3^{-\frac{2}{3}}$$

Subtract 4\sqrt{7} from both sides.

$$-7\times 3^{-\frac{2}{3}}x^{2}+3\sqrt{7}x^{2}=21\times 3^{-\frac{2}{3}}-4\sqrt{7}$$

Combine all terms containing x.

$$\left(-7\times 3^{-\frac{2}{3}}+3\sqrt{7}\right)x^{2}=21\times 3^{-\frac{2}{3}}-4\sqrt{7}$$

Dividing by -7\times 3^{-\frac{2}{3}}+3\sqrt{7} undoes the multiplication by -7\times 3^{-\frac{2}{3}}+3\sqrt{7}.

$$x^{2}=\frac{\frac{21}{3^{\frac{2}{3}}}-4\sqrt{7}}{-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}}$$

Divide -4\sqrt{7}+\frac{21}{3^{\frac{2}{3}}} by -7\times 3^{-\frac{2}{3}}+3\sqrt{7}.

$$x^{2}=\frac{\sqrt{7}\left(\sqrt{7}+3^{\frac{5}{3}}\right)\left(-4\sqrt{7}\times 3^{\frac{2}{3}}+21\right)}{7\left(9\times 3^{\frac{4}{3}}-7\right)}$$

Take the square root of both sides of the equation.

$$x=\frac{i\sqrt{-\left(\sqrt{7}+3^{\frac{5}{3}}\right)\left(-4\sqrt{7}\times 3^{\frac{2}{3}}+21\right)}}{\sqrt[4]{7}\sqrt{3^{\frac{10}{3}}-7}} x=-\frac{i\sqrt{-\left(\sqrt{7}+3^{\frac{5}{3}}\right)\left(-4\sqrt{7}\times 3^{\frac{2}{3}}+21\right)}}{\sqrt[4]{7}\sqrt{3^{\frac{10}{3}}-7}}$$

The equation is now solved.

$$x=\frac{i\sqrt{\left(\sqrt{7}+3^{\frac{5}{3}}\right)\left(4\sqrt{7}\times 3^{\frac{2}{3}}-21\right)}}{\sqrt[4]{7}\sqrt{3^{\frac{10}{3}}-7}} x=-\frac{i\sqrt{\left(\sqrt{7}+3^{\frac{5}{3}}\right)\left(4\sqrt{7}\times 3^{\frac{2}{3}}-21\right)}}{\sqrt[4]{7}\sqrt{3^{\frac{10}{3}}-7}}$$

Steps Using the Quadratic Formula

Rationalize the denominator of \frac{4+3x^{2}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\left(4+3x^{2}\right)\sqrt{7}}{\left(\sqrt{7}\right)^{2}}=\frac{3+x^{2}}{\sqrt[3]{9}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(4+3x^{2}\right)\sqrt{7}}{7}=\frac{3+x^{2}}{\sqrt[3]{9}}$$

Use the distributive property to multiply 4+3x^{2} by \sqrt{7}.

$$\frac{4\sqrt{7}+3x^{2}\sqrt{7}}{7}=\frac{3+x^{2}}{\sqrt[3]{9}}$$

Subtract \frac{3+x^{2}}{\sqrt[3]{9}} from both sides.

$$\frac{4\sqrt{7}+3x^{2}\sqrt{7}}{7}-\frac{3+x^{2}}{\sqrt[3]{9}}=0$$

Multiply both sides of the equation by 7.

$$4\sqrt{7}+3x^{2}\sqrt{7}-7\times 3^{-\frac{2}{3}}\left(3+x^{2}\right)=0$$

Reorder the terms.

$$-7\times 3^{-\frac{2}{3}}\left(x^{2}+3\right)+3\sqrt{7}x^{2}+4\sqrt{7}=0$$

Use the distributive property to multiply -7\times 3^{-\frac{2}{3}} by x^{2}+3.

$$-7\times 3^{-\frac{2}{3}}x^{2}-21\times 3^{-\frac{2}{3}}+3\sqrt{7}x^{2}+4\sqrt{7}=0$$

Combine all terms containing x.

$$\left(-7\times 3^{-\frac{2}{3}}+3\sqrt{7}\right)x^{2}-21\times 3^{-\frac{2}{3}}+4\sqrt{7}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)x^{2}-\frac{21}{3^{\frac{2}{3}}}+4\sqrt{7}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7\times 3^{-\frac{2}{3}}+3\sqrt{7} for a, 0 for b, and 4\sqrt{7}-\frac{21}{3^{\frac{2}{3}}} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)\left(-\frac{21}{3^{\frac{2}{3}}}+4\sqrt{7}\right)}}{2\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)\left(-\frac{21}{3^{\frac{2}{3}}}+4\sqrt{7}\right)}}{2\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)}$$

Multiply -4 times -7\times 3^{-\frac{2}{3}}+3\sqrt{7}.

$$x=\frac{0±\sqrt{\left(\frac{28}{3^{\frac{2}{3}}}-12\sqrt{7}\right)\left(-\frac{21}{3^{\frac{2}{3}}}+4\sqrt{7}\right)}}{2\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)}$$

Multiply -12\sqrt{7}+\frac{28}{3^{\frac{2}{3}}} times 4\sqrt{7}-\frac{21}{3^{\frac{2}{3}}}.

$$x=\frac{0±\sqrt{\frac{364\sqrt{7}}{3^{\frac{2}{3}}}-\frac{588}{3^{\frac{4}{3}}}-336}}{2\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)}$$

Take the square root of -336+\frac{364\sqrt{7}\sqrt[3]{3}}{3}-\frac{196\times 3^{\frac{2}{3}}}{3}.

$$x=\frac{0±\frac{2i\sqrt{-273\sqrt{7}\sqrt[3]{3}+147\times 3^{\frac{2}{3}}+756}}{3}}{2\left(-\frac{7}{3^{\frac{2}{3}}}+3\sqrt{7}\right)}$$

Multiply 2 times -7\times 3^{-\frac{2}{3}}+3\sqrt{7}.

$$x=\frac{0±\frac{2i\sqrt{-273\sqrt{7}\sqrt[3]{3}+147\times 3^{\frac{2}{3}}+756}}{3}}{-\frac{14}{3^{\frac{2}{3}}}+6\sqrt{7}}$$

Now solve the equation x=\frac{0±\frac{2i\sqrt{-273\sqrt{7}\sqrt[3]{3}+147\times 3^{\frac{2}{3}}+756}}{3}}{-\frac{14}{3^{\frac{2}{3}}}+6\sqrt{7}} when ± is plus.

$$x=\frac{3^{\frac{7}{6}}\left(\sqrt{7}+3^{\frac{5}{3}}\right)i\sqrt{-13\sqrt{7}\sqrt[3]{3}+7\times 3^{\frac{2}{3}}+36}}{3\left(9\times 3^{\frac{4}{3}}-7\right)}$$

Now solve the equation x=\frac{0±\frac{2i\sqrt{-273\sqrt{7}\sqrt[3]{3}+147\times 3^{\frac{2}{3}}+756}}{3}}{-\frac{14}{3^{\frac{2}{3}}}+6\sqrt{7}} when ± is minus.

$$x=-\frac{3^{\frac{7}{6}}\left(\sqrt{7}+3^{\frac{5}{3}}\right)i\sqrt{-13\sqrt{7}\sqrt[3]{3}+7\times 3^{\frac{2}{3}}+36}}{3\left(9\times 3^{\frac{4}{3}}-7\right)}$$

The equation is now solved.

$$x=\frac{3^{\frac{7}{6}}\left(\sqrt{7}+3^{\frac{5}{3}}\right)i\sqrt{-13\sqrt{7}\sqrt[3]{3}+7\times 3^{\frac{2}{3}}+36}}{3\left(9\times 3^{\frac{4}{3}}-7\right)} x=-\frac{3^{\frac{7}{6}}\left(\sqrt{7}+3^{\frac{5}{3}}\right)i\sqrt{-13\sqrt{7}\sqrt[3]{3}+7\times 3^{\frac{2}{3}}+36}}{3\left(9\times 3^{\frac{4}{3}}-7\right)}$$