Solve xsqrt{36-x^2}+xsqrt{25-x^2}=24

$$x \sqrt{ 36- { x }^{ 2 } } +x \sqrt{ 25- { x }^{ 2 } } =24$$

Solve for x

$$x = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

$$x = \frac{24 \sqrt{97}}{97} \approx 2.436830796$$

Solution Steps

Subtract x\sqrt{25-x^{2}} from both sides of the equation.

$$x\sqrt{36-x^{2}}=24-x\sqrt{25-x^{2}}$$

Reorder the terms.

$$x\sqrt{-x^{2}+36}=-x\sqrt{-x^{2}+25}+24$$

Square both sides of the equation.

$$\left(x\sqrt{-x^{2}+36}\right)^{2}=\left(-x\sqrt{-x^{2}+25}+24\right)^{2}$$

Expand \left(x\sqrt{-x^{2}+36}\right)^{2}.

$$x^{2}\left(\sqrt{-x^{2}+36}\right)^{2}=\left(-x\sqrt{-x^{2}+25}+24\right)^{2}$$

Calculate \sqrt{-x^{2}+36} to the power of 2 and get -x^{2}+36.

$$x^{2}\left(-x^{2}+36\right)=\left(-x\sqrt{-x^{2}+25}+24\right)^{2}$$

Use the distributive property to multiply x^{2} by -x^{2}+36.

$$-x^{4}+36x^{2}=\left(-x\sqrt{-x^{2}+25}+24\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-x\sqrt{-x^{2}+25}+24\right)^{2}.

$$-x^{4}+36x^{2}=x^{2}\left(\sqrt{-x^{2}+25}\right)^{2}-48x\sqrt{-x^{2}+25}+576$$

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

$$-x^{4}+36x^{2}=x^{2}\left(-x^{2}+25\right)-48x\sqrt{-x^{2}+25}+576$$

Use the distributive property to multiply x^{2} by -x^{2}+25.

$$-x^{4}+36x^{2}=-x^{4}+25x^{2}-48x\sqrt{-x^{2}+25}+576$$

Subtract -x^{4}+25x^{2}+576 from both sides of the equation.

$$-x^{4}+36x^{2}-\left(-x^{4}+25x^{2}+576\right)=-48x\sqrt{-x^{2}+25}$$

To find the opposite of -x^{4}+25x^{2}+576, find the opposite of each term.

$$-x^{4}+36x^{2}+x^{4}-25x^{2}-576=-48x\sqrt{-x^{2}+25}$$

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

$$36x^{2}-25x^{2}-576=-48x\sqrt{-x^{2}+25}$$

Combine 36x^{2} and -25x^{2} to get 11x^{2}.

$$11x^{2}-576=-48x\sqrt{-x^{2}+25}$$

Square both sides of the equation.

$$\left(11x^{2}-576\right)^{2}=\left(-48x\sqrt{-x^{2}+25}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11x^{2}-576\right)^{2}.

$$121\left(x^{2}\right)^{2}-12672x^{2}+331776=\left(-48x\sqrt{-x^{2}+25}\right)^{2}$$

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

$$121x^{4}-12672x^{2}+331776=\left(-48x\sqrt{-x^{2}+25}\right)^{2}$$

Expand \left(-48x\sqrt{-x^{2}+25}\right)^{2}.

$$121x^{4}-12672x^{2}+331776=\left(-48\right)^{2}x^{2}\left(\sqrt{-x^{2}+25}\right)^{2}$$

Calculate -48 to the power of 2 and get 2304.

$$121x^{4}-12672x^{2}+331776=2304x^{2}\left(\sqrt{-x^{2}+25}\right)^{2}$$

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

$$121x^{4}-12672x^{2}+331776=2304x^{2}\left(-x^{2}+25\right)$$

Use the distributive property to multiply 2304x^{2} by -x^{2}+25.

$$121x^{4}-12672x^{2}+331776=-2304x^{4}+57600x^{2}$$

Add 2304x^{4} to both sides.

$$121x^{4}-12672x^{2}+331776+2304x^{4}=57600x^{2}$$

Combine 121x^{4} and 2304x^{4} to get 2425x^{4}.

$$2425x^{4}-12672x^{2}+331776=57600x^{2}$$

Subtract 57600x^{2} from both sides.

$$2425x^{4}-12672x^{2}[phone]x^{2}=0$$

Combine -12672x^{2} and -57600x^{2} to get -70272x^{2}.

$$2425x^{4}-70272x^{2}+331776=0$$

Substitute t for x^{2}.

$$2425t^{2}-70272t+331776=0$$

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 2425 for a, -70272 for b, and 331776 for c in the quadratic formula.

$$t=\frac{-\left(-70272\right)±\sqrt{\left(-70272\right)^{2}-4\times 2425\times 331776}}{2\times 2425}$$

Do the calculations.

$$t=\frac{70272±41472}{4850}$$

Solve the equation t=\frac{70272±41472}{4850} when ± is plus and when ± is minus.

$$t=\frac{576}{25} t=\frac{576}{97}$$

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

$$x=\frac{24}{5} x=-\frac{24}{5} x=\frac{24\sqrt{97}}{97} x=-\frac{24\sqrt{97}}{97}$$

Substitute \frac{24}{5} for x in the equation x\sqrt{36-x^{2}}+x\sqrt{25-x^{2}}=24.

$$\frac{24}{5}\sqrt{36-\left(\frac{24}{5}\right)^{2}}+\frac{24}{5}\sqrt{25-\left(\frac{24}{5}\right)^{2}}=24$$

Simplify. The value x=\frac{24}{5} satisfies the equation.

$$24=24$$

Substitute -\frac{24}{5} for x in the equation x\sqrt{36-x^{2}}+x\sqrt{25-x^{2}}=24.

$$-\frac{24}{5}\sqrt{36-\left(-\frac{24}{5}\right)^{2}}-\frac{24}{5}\sqrt{25-\left(-\frac{24}{5}\right)^{2}}=24$$

Simplify. The value x=-\frac{24}{5} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-24=24$$

Substitute \frac{24\sqrt{97}}{97} for x in the equation x\sqrt{36-x^{2}}+x\sqrt{25-x^{2}}=24.

$$\frac{24\sqrt{97}}{97}\sqrt{36-\left(\frac{24\sqrt{97}}{97}\right)^{2}}+\frac{24\sqrt{97}}{97}\sqrt{25-\left(\frac{24\sqrt{97}}{97}\right)^{2}}=24$$

Simplify. The value x=\frac{24\sqrt{97}}{97} satisfies the equation.

$$24=24$$

Substitute -\frac{24\sqrt{97}}{97} for x in the equation x\sqrt{36-x^{2}}+x\sqrt{25-x^{2}}=24.

$$\left(-\frac{24\sqrt{97}}{97}\right)\sqrt{36-\left(-\frac{24\sqrt{97}}{97}\right)^{2}}+\left(-\frac{24\sqrt{97}}{97}\right)\sqrt{25-\left(-\frac{24\sqrt{97}}{97}\right)^{2}}=24$$

Simplify. The value x=-\frac{24\sqrt{97}}{97} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-24=24$$

Substitute \frac{24}{5} for x in the equation x\sqrt{36-x^{2}}+x\sqrt{25-x^{2}}=24.

$$\frac{24}{5}\sqrt{36-\left(\frac{24}{5}\right)^{2}}+\frac{24}{5}\sqrt{25-\left(\frac{24}{5}\right)^{2}}=24$$

Simplify. The value x=\frac{24}{5} satisfies the equation.

$$24=24$$

Substitute \frac{24\sqrt{97}}{97} for x in the equation x\sqrt{36-x^{2}}+x\sqrt{25-x^{2}}=24.

$$\frac{24\sqrt{97}}{97}\sqrt{36-\left(\frac{24\sqrt{97}}{97}\right)^{2}}+\frac{24\sqrt{97}}{97}\sqrt{25-\left(\frac{24\sqrt{97}}{97}\right)^{2}}=24$$

Simplify. The value x=\frac{24\sqrt{97}}{97} satisfies the equation.

$$24=24$$

List all solutions of x\sqrt{-x^{2}+36}=-x\sqrt{-x^{2}+25}+24.

$$x=\frac{24}{5} x=\frac{24\sqrt{97}}{97}$$