Solve 102400=(900-x^2)(x^2-10)

$$102400=(900- { x }^{ 2 } )( { x }^{ 2 } -10)$$

Solve for x

$$x=-\sqrt{455-75\sqrt{17}}\approx -12.073403749$$

$$x=\sqrt{455-75\sqrt{17}}\approx 12.073403749$$

$$x=\sqrt{75\sqrt{17}+455}\approx 27.644763011$$

$$x=-\sqrt{75\sqrt{17}+455}\approx -27.644763011$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 900-x^{2} by x^{2}-10 and combine like terms.

$$102400=910x^{2}-9000-x^{4}$$

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

$$910x^{2}-9000-x^{4}=102400$$

Subtract 102400 from both sides.

$$910x^{2}-9000-x^{4}-102400=0$$

Subtract 102400 from -9000 to get -111400.

$$910x^{2}-111400-x^{4}=0$$

Substitute t for x^{2}.

$$-t^{2}+910t-111400=0$$

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 -1 for a, 910 for b, and -111400 for c in the quadratic formula.

$$t=\frac{-910±\sqrt{910^{2}-4\left(-1\right)\left(-111400\right)}}{-2}$$

Do the calculations.

$$t=\frac{-910±150\sqrt{17}}{-2}$$

Solve the equation t=\frac{-910±150\sqrt{17}}{-2} when ± is plus and when ± is minus.

$$t=455-75\sqrt{17} t=75\sqrt{17}+455$$

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

$$x=\sqrt{455-75\sqrt{17}} x=-\sqrt{455-75\sqrt{17}} x=\sqrt{75\sqrt{17}+455} x=-\sqrt{75\sqrt{17}+455}$$