Solve (6^2-3x^2)(6^2-x^2)=945

$$( { 6 }^{ 2 } -3 { x }^{ 2 } )( { 6 }^{ 2 } - { x }^{ 2 } )=945$$

Solve for x

$$x=-\sqrt{24-3\sqrt{51}}\approx -1.604903335$$

$$x=\sqrt{24-3\sqrt{51}}\approx 1.604903335$$

$$x=\sqrt{3\sqrt{51}+24}\approx 6.739754097$$

$$x=-\sqrt{3\sqrt{51}+24}\approx -6.739754097$$

Steps Using the Quadratic Formula

Calculate 6 to the power of 2 and get 36.

$$\left(36-3x^{2}\right)\left(6^{2}-x^{2}\right)=945$$

Calculate 6 to the power of 2 and get 36.

$$\left(36-3x^{2}\right)\left(36-x^{2}\right)=945$$

Use the distributive property to multiply 36-3x^{2} by 36-x^{2} and combine like terms.

$$1296-144x^{2}+3x^{4}=945$$

Subtract 945 from both sides.

$$1296-144x^{2}+3x^{4}-945=0$$

Subtract 945 from 1296 to get 351.

$$351-144x^{2}+3x^{4}=0$$

Substitute t for x^{2}.

$$3t^{2}-144t+351=0$$

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 3 for a, -144 for b, and 351 for c in the quadratic formula.

$$t=\frac{-\left(-144\right)±\sqrt{\left(-144\right)^{2}-4\times 3\times 351}}{2\times 3}$$

Do the calculations.

$$t=\frac{144±18\sqrt{51}}{6}$$

Solve the equation t=\frac{144±18\sqrt{51}}{6} when ± is plus and when ± is minus.

$$t=3\sqrt{51}+24 t=24-3\sqrt{51}$$

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

$$x=\sqrt{3\sqrt{51}+24} x=-\sqrt{3\sqrt{51}+24} x=\sqrt{24-3\sqrt{51}} x=-\sqrt{24-3\sqrt{51}}$$