Solve sqrt{x+5}+2=sqrt{2x+17}

$$\sqrt { x + 5 } + 2 = \sqrt { 2 x + 17 }$$

Solve for x

$$x=4$$

$$x=-4$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+5}+2\right)^{2}=\left(\sqrt{2x+17}\right)^{2}$$

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

$$\left(\sqrt{x+5}\right)^{2}+4\sqrt{x+5}+4=\left(\sqrt{2x+17}\right)^{2}$$

Calculate \sqrt{x+5} to the power of 2 and get x+5.

$$x+5+4\sqrt{x+5}+4=\left(\sqrt{2x+17}\right)^{2}$$

Add 5 and 4 to get 9.

$$x+9+4\sqrt{x+5}=\left(\sqrt{2x+17}\right)^{2}$$

Calculate \sqrt{2x+17} to the power of 2 and get 2x+17.

$$x+9+4\sqrt{x+5}=2x+17$$

Subtract x+9 from both sides of the equation.

$$4\sqrt{x+5}=2x+17-\left(x+9\right)$$

To find the opposite of x+9, find the opposite of each term.

$$4\sqrt{x+5}=2x+17-x-9$$

Combine 2x and -x to get x.

$$4\sqrt{x+5}=x+17-9$$

Subtract 9 from 17 to get 8.

$$4\sqrt{x+5}=x+8$$

Square both sides of the equation.

$$\left(4\sqrt{x+5}\right)^{2}=\left(x+8\right)^{2}$$

Expand \left(4\sqrt{x+5}\right)^{2}.

$$4^{2}\left(\sqrt{x+5}\right)^{2}=\left(x+8\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16\left(\sqrt{x+5}\right)^{2}=\left(x+8\right)^{2}$$

Calculate \sqrt{x+5} to the power of 2 and get x+5.

$$16\left(x+5\right)=\left(x+8\right)^{2}$$

Use the distributive property to multiply 16 by x+5.

$$16x+80=\left(x+8\right)^{2}$$

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

$$16x+80=x^{2}+16x+64$$

Subtract x^{2} from both sides.

$$16x+80-x^{2}=16x+64$$

Subtract 16x from both sides.

$$16x+80-x^{2}-16x=64$$

Combine 16x and -16x to get 0.

$$80-x^{2}=64$$

Subtract 80 from both sides.

$$-x^{2}=64-80$$

Subtract 80 from 64 to get -16.

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

Divide both sides by -1.

$$x^{2}=\frac{-16}{-1}$$

Fraction \frac{-16}{-1} can be simplified to 16 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=16$$

Take the square root of both sides of the equation.

$$x=4 x=-4$$

Substitute 4 for x in the equation \sqrt{x+5}+2=\sqrt{2x+17}.

$$\sqrt{4+5}+2=\sqrt{2\times 4+17}$$

Simplify. The value x=4 satisfies the equation.

$$5=5$$

Substitute -4 for x in the equation \sqrt{x+5}+2=\sqrt{2x+17}.

$$\sqrt{-4+5}+2=\sqrt{2\left(-4\right)+17}$$

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

$$3=3$$

List all solutions of \sqrt{x+5}+2=\sqrt{2x+17}.

$$x=4 x=-4$$