Solve sqrt{2x+9}+sqrt{1-2x}=4

$$\sqrt { 2 x + 9 } + \sqrt { 1 - 2 x } = 4$$

Solve for x

$$x=-4$$

$$x=0$$

Solution Steps

Subtract \sqrt{1-2x} from both sides of the equation.

$$\sqrt{2x+9}=4-\sqrt{1-2x}$$

Square both sides of the equation.

$$\left(\sqrt{2x+9}\right)^{2}=\left(4-\sqrt{1-2x}\right)^{2}$$

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

$$2x+9=\left(4-\sqrt{1-2x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{1-2x}\right)^{2}.

$$2x+9=16-8\sqrt{1-2x}+\left(\sqrt{1-2x}\right)^{2}$$

Calculate \sqrt{1-2x} to the power of 2 and get 1-2x.

$$2x+9=16-8\sqrt{1-2x}+1-2x$$

Add 16 and 1 to get 17.

$$2x+9=17-8\sqrt{1-2x}-2x$$

Subtract 17-2x from both sides of the equation.

$$2x+9-\left(17-2x\right)=-8\sqrt{1-2x}$$

To find the opposite of 17-2x, find the opposite of each term.

$$2x+9-17+2x=-8\sqrt{1-2x}$$

Subtract 17 from 9 to get -8.

$$2x-8+2x=-8\sqrt{1-2x}$$

Combine 2x and 2x to get 4x.

$$4x-8=-8\sqrt{1-2x}$$

Square both sides of the equation.

$$\left(4x-8\right)^{2}=\left(-8\sqrt{1-2x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-8\right)^{2}.

$$16x^{2}-64x+64=\left(-8\sqrt{1-2x}\right)^{2}$$

Expand \left(-8\sqrt{1-2x}\right)^{2}.

$$16x^{2}-64x+64=\left(-8\right)^{2}\left(\sqrt{1-2x}\right)^{2}$$

Calculate -8 to the power of 2 and get 64.

$$16x^{2}-64x+64=64\left(\sqrt{1-2x}\right)^{2}$$

Calculate \sqrt{1-2x} to the power of 2 and get 1-2x.

$$16x^{2}-64x+64=64\left(1-2x\right)$$

Use the distributive property to multiply 64 by 1-2x.

$$16x^{2}-64x+64=64-128x$$

Subtract 64 from both sides.

$$16x^{2}-64x+64-64=-128x$$

Subtract 64 from 64 to get 0.

$$16x^{2}-64x=-128x$$

Add 128x to both sides.

$$16x^{2}-64x+128x=0$$

Combine -64x and 128x to get 64x.

$$16x^{2}+64x=0$$

Factor out x.

$$x\left(16x+64\right)=0$$

To find equation solutions, solve x=0 and 16x+64=0.

$$x=0 x=-4$$

Substitute 0 for x in the equation \sqrt{2x+9}+\sqrt{1-2x}=4.

$$\sqrt{2\times 0+9}+\sqrt{1-2\times 0}=4$$

Simplify. The value x=0 satisfies the equation.

$$4=4$$

Substitute -4 for x in the equation \sqrt{2x+9}+\sqrt{1-2x}=4.

$$\sqrt{2\left(-4\right)+9}+\sqrt{1-2\left(-4\right)}=4$$

Simplify. The value x=-4 satisfies the equation.

$$4=4$$

List all solutions of \sqrt{2x+9}=-\sqrt{1-2x}+4.

$$x=0 x=-4$$