Solve sqrt{4-x}+sqrt{9+x}=5

$$\sqrt { 4 - x } + \sqrt { 9 + x } = 5$$

Solve for x

$$x=-5$$

$$x=0$$

Solution Steps

Subtract \sqrt{9+x} from both sides of the equation.

$$\sqrt{4-x}=5-\sqrt{9+x}$$

Square both sides of the equation.

$$\left(\sqrt{4-x}\right)^{2}=\left(5-\sqrt{9+x}\right)^{2}$$

Calculate \sqrt{4-x} to the power of 2 and get 4-x.

$$4-x=\left(5-\sqrt{9+x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-\sqrt{9+x}\right)^{2}.

$$4-x=25-10\sqrt{9+x}+\left(\sqrt{9+x}\right)^{2}$$

Calculate \sqrt{9+x} to the power of 2 and get 9+x.

$$4-x=25-10\sqrt{9+x}+9+x$$

Add 25 and 9 to get 34.

$$4-x=34-10\sqrt{9+x}+x$$

Subtract 34+x from both sides of the equation.

$$4-x-\left(34+x\right)=-10\sqrt{9+x}$$

To find the opposite of 34+x, find the opposite of each term.

$$4-x-34-x=-10\sqrt{9+x}$$

Subtract 34 from 4 to get -30.

$$-30-x-x=-10\sqrt{9+x}$$

Combine -x and -x to get -2x.

$$-30-2x=-10\sqrt{9+x}$$

Square both sides of the equation.

$$\left(-30-2x\right)^{2}=\left(-10\sqrt{9+x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-30-2x\right)^{2}.

$$900+120x+4x^{2}=\left(-10\sqrt{9+x}\right)^{2}$$

Expand \left(-10\sqrt{9+x}\right)^{2}.

$$900+120x+4x^{2}=\left(-10\right)^{2}\left(\sqrt{9+x}\right)^{2}$$

Calculate -10 to the power of 2 and get 100.

$$900+120x+4x^{2}=100\left(\sqrt{9+x}\right)^{2}$$

Calculate \sqrt{9+x} to the power of 2 and get 9+x.

$$900+120x+4x^{2}=100\left(9+x\right)$$

Use the distributive property to multiply 100 by 9+x.

$$900+120x+4x^{2}=900+100x$$

Subtract 900 from both sides.

$$900+120x+4x^{2}-900=100x$$

Subtract 900 from 900 to get 0.

$$120x+4x^{2}=100x$$

Subtract 100x from both sides.

$$120x+4x^{2}-100x=0$$

Combine 120x and -100x to get 20x.

$$20x+4x^{2}=0$$

Factor out x.

$$x\left(20+4x\right)=0$$

To find equation solutions, solve x=0 and 20+4x=0.

$$x=0 x=-5$$

Substitute 0 for x in the equation \sqrt{4-x}+\sqrt{9+x}=5.

$$\sqrt{4-0}+\sqrt{9+0}=5$$

Simplify. The value x=0 satisfies the equation.

$$5=5$$

Substitute -5 for x in the equation \sqrt{4-x}+\sqrt{9+x}=5.

$$\sqrt{4-\left(-5\right)}+\sqrt{9-5}=5$$

Simplify. The value x=-5 satisfies the equation.

$$5=5$$

List all solutions of \sqrt{4-x}=-\sqrt{x+9}+5.

$$x=0 x=-5$$