Solve 1296=(k^2+4)(100+25k^2)

$$1296 = ( k ^ { 2 } + 4 ) ( 100 + 25 k ^ { 2 } )$$

Solve for k

$$k = -\frac{4 \sqrt{5}}{5} \approx -1.788854382$$

$$k = \frac{4 \sqrt{5}}{5} \approx 1.788854382$$

Steps Using the Quadratic Formula

Use the distributive property to multiply k^{2}+4 by 100+25k^{2} and combine like terms.

$$1296=200k^{2}+25k^{4}+400$$

Swap sides so that all variable terms are on the left hand side.

$$200k^{2}+25k^{4}+400=1296$$

Subtract 1296 from both sides.

$$200k^{2}+25k^{4}+400-1296=0$$

Subtract 1296 from 400 to get -896.

$$200k^{2}+25k^{4}-896=0$$

Substitute t for k^{2}.

$$25t^{2}+200t-896=0$$

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 25 for a, 200 for b, and -896 for c in the quadratic formula.

$$t=\frac{-200±\sqrt{200^{2}-4\times 25\left(-896\right)}}{2\times 25}$$

Do the calculations.

$$t=\frac{-200±360}{50}$$

Solve the equation t=\frac{-200±360}{50} when ± is plus and when ± is minus.

$$t=\frac{16}{5} t=-\frac{56}{5}$$

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

$$k=\frac{4\sqrt{5}}{5} k=-\frac{4\sqrt{5}}{5}$$