Solve 16k^2+24k^4/(2k^2+1)^2=20/9

$$\frac { 16 k ^ { 2 } + 24 k ^ { 4 } } { ( 2 k ^ { 2 } + 1 ) ^ { 2 } } = \frac { 20 } { 9 }$$

Solve for k

$$k=-\frac{\sqrt{102\sqrt{26}-272}}{34}\approx -0.463270298$$

$$k=\frac{\sqrt{102\sqrt{26}-272}}{34}\approx 0.463270298$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 9\left(2k^{2}+1\right)^{2}, the least common multiple of \left(2k^{2}+1\right)^{2},9.

$$9\left(16k^{2}+24k^{4}\right)=20\left(2k^{2}+1\right)^{2}$$

Use the distributive property to multiply 9 by 16k^{2}+24k^{4}.

$$144k^{2}+216k^{4}=20\left(2k^{2}+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2k^{2}+1\right)^{2}.

$$144k^{2}+216k^{4}=20\left(4\left(k^{2}\right)^{2}+4k^{2}+1\right)$$

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

$$144k^{2}+216k^{4}=20\left(4k^{4}+4k^{2}+1\right)$$

Use the distributive property to multiply 20 by 4k^{4}+4k^{2}+1.

$$144k^{2}+216k^{4}=80k^{4}+80k^{2}+20$$

Subtract 80k^{4} from both sides.

$$144k^{2}+216k^{4}-80k^{4}=80k^{2}+20$$

Combine 216k^{4} and -80k^{4} to get 136k^{4}.

$$144k^{2}+136k^{4}=80k^{2}+20$$

Subtract 80k^{2} from both sides.

$$144k^{2}+136k^{4}-80k^{2}=20$$

Combine 144k^{2} and -80k^{2} to get 64k^{2}.

$$64k^{2}+136k^{4}=20$$

Subtract 20 from both sides.

$$64k^{2}+136k^{4}-20=0$$

Substitute t for k^{2}.

$$136t^{2}+64t-20=0$$

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 136 for a, 64 for b, and -20 for c in the quadratic formula.

$$t=\frac{-64±\sqrt{64^{2}-4\times 136\left(-20\right)}}{2\times 136}$$

Do the calculations.

$$t=\frac{-64±24\sqrt{26}}{272}$$

Solve the equation t=\frac{-64±24\sqrt{26}}{272} when ± is plus and when ± is minus.

$$t=\frac{3\sqrt{26}}{34}-\frac{4}{17} t=-\frac{3\sqrt{26}}{34}-\frac{4}{17}$$

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

$$k=\frac{\sqrt{\frac{6\sqrt{26}-16}{17}}}{2} k=-\frac{\sqrt{\frac{6\sqrt{26}-16}{17}}}{2}$$