Solve 144+25=(sqrt{25-x^2}+sqrt{144-x^2})^2

$$144 + 25 = ( \sqrt { 25 - x ^ { 2 } } + \sqrt { 144 - x ^ { 2 } } ) ^ { 2 }$$

Solve for x

$$x = \frac{60}{13} = 4\frac{8}{13} \approx 4.615384615$$

$$x = -\frac{60}{13} = -4\frac{8}{13} \approx -4.615384615$$

Solution Steps

Add 144 and 25 to get 169.

$$169=\left(\sqrt{25-x^{2}}+\sqrt{144-x^{2}}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{25-x^{2}}+\sqrt{144-x^{2}}\right)^{2}.

$$169=\left(\sqrt{25-x^{2}}\right)^{2}+2\sqrt{25-x^{2}}\sqrt{144-x^{2}}+\left(\sqrt{144-x^{2}}\right)^{2}$$

Calculate \sqrt{25-x^{2}} to the power of 2 and get 25-x^{2}.

$$169=25-x^{2}+2\sqrt{25-x^{2}}\sqrt{144-x^{2}}+\left(\sqrt{144-x^{2}}\right)^{2}$$

Calculate \sqrt{144-x^{2}} to the power of 2 and get 144-x^{2}.

$$169=25-x^{2}+2\sqrt{25-x^{2}}\sqrt{144-x^{2}}+144-x^{2}$$

Add 25 and 144 to get 169.

$$169=169-x^{2}+2\sqrt{25-x^{2}}\sqrt{144-x^{2}}-x^{2}$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$169=169-2x^{2}+2\sqrt{25-x^{2}}\sqrt{144-x^{2}}$$

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

$$169-2x^{2}+2\sqrt{25-x^{2}}\sqrt{144-x^{2}}=169$$

Subtract 169-2x^{2} from both sides of the equation.

$$2\sqrt{25-x^{2}}\sqrt{144-x^{2}}=169-\left(169-2x^{2}\right)$$

To find the opposite of 169-2x^{2}, find the opposite of each term.

$$2\sqrt{25-x^{2}}\sqrt{144-x^{2}}=169-169+2x^{2}$$

Subtract 169 from 169 to get 0.

$$2\sqrt{25-x^{2}}\sqrt{144-x^{2}}=2x^{2}$$

Cancel out 2 on both sides.

$$\sqrt{25-x^{2}}\sqrt{144-x^{2}}=x^{2}$$

Square both sides of the equation.

$$\left(\sqrt{25-x^{2}}\sqrt{144-x^{2}}\right)^{2}=\left(x^{2}\right)^{2}$$

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

$$\left(\sqrt{25-x^{2}}\sqrt{144-x^{2}}\right)^{2}=x^{4}$$

Expand \left(\sqrt{25-x^{2}}\sqrt{144-x^{2}}\right)^{2}.

$$\left(\sqrt{25-x^{2}}\right)^{2}\left(\sqrt{144-x^{2}}\right)^{2}=x^{4}$$

Calculate \sqrt{25-x^{2}} to the power of 2 and get 25-x^{2}.

$$\left(25-x^{2}\right)\left(\sqrt{144-x^{2}}\right)^{2}=x^{4}$$

Calculate \sqrt{144-x^{2}} to the power of 2 and get 144-x^{2}.

$$\left(25-x^{2}\right)\left(144-x^{2}\right)=x^{4}$$

Use the distributive property to multiply 25-x^{2} by 144-x^{2} and combine like terms.

$$3600-169x^{2}+x^{4}=x^{4}$$

Subtract x^{4} from both sides.

$$3600-169x^{2}+x^{4}-x^{4}=0$$

Combine x^{4} and -x^{4} to get 0.

$$3600-169x^{2}=0$$

Subtract 3600 from both sides. Anything subtracted from zero gives its negation.

$$-169x^{2}=-3600$$

Divide both sides by -169.

$$x^{2}=\frac{-3600}{-169}$$

Fraction \frac{-3600}{-169} can be simplified to \frac{3600}{169} by removing the negative sign from both the numerator and the denominator.

$$x^{2}=\frac{3600}{169}$$

Take the square root of both sides of the equation.

$$x=\frac{60}{13} x=-\frac{60}{13}$$

Substitute \frac{60}{13} for x in the equation 144+25=\left(\sqrt{25-x^{2}}+\sqrt{144-x^{2}}\right)^{2}.

$$144+25=\left(\sqrt{25-\left(\frac{60}{13}\right)^{2}}+\sqrt{144-\left(\frac{60}{13}\right)^{2}}\right)^{2}$$

Simplify. The value x=\frac{60}{13} satisfies the equation.

$$169=169$$

Substitute -\frac{60}{13} for x in the equation 144+25=\left(\sqrt{25-x^{2}}+\sqrt{144-x^{2}}\right)^{2}.

$$144+25=\left(\sqrt{25-\left(-\frac{60}{13}\right)^{2}}+\sqrt{144-\left(-\frac{60}{13}\right)^{2}}\right)^{2}$$

Simplify. The value x=-\frac{60}{13} satisfies the equation.

$$169=169$$

List all solutions of \sqrt{25-x^{2}}\sqrt{144-x^{2}}=x^{2}.

$$x=\frac{60}{13} x=-\frac{60}{13}$$