Solve sqrt{w+2}+sqrt{w-23}=5

$$\sqrt { w + 2 } + \sqrt { w - 23 } = 5$$

Solve for w

$$w=23$$

Solution Steps

Subtract \sqrt{w-23} from both sides of the equation.

$$\sqrt{w+2}=5-\sqrt{w-23}$$

Square both sides of the equation.

$$\left(\sqrt{w+2}\right)^{2}=\left(5-\sqrt{w-23}\right)^{2}$$

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

$$w+2=\left(5-\sqrt{w-23}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-\sqrt{w-23}\right)^{2}.

$$w+2=25-10\sqrt{w-23}+\left(\sqrt{w-23}\right)^{2}$$

Calculate \sqrt{w-23} to the power of 2 and get w-23.

$$w+2=25-10\sqrt{w-23}+w-23$$

Subtract 23 from 25 to get 2.

$$w+2=2-10\sqrt{w-23}+w$$

Add 10\sqrt{w-23} to both sides.

$$w+2+10\sqrt{w-23}=2+w$$

Subtract w from both sides.

$$w+2+10\sqrt{w-23}-w=2$$

Combine w and -w to get 0.

$$2+10\sqrt{w-23}=2$$

Subtract 2 from both sides.

$$10\sqrt{w-23}=2-2$$

Subtract 2 from 2 to get 0.

$$10\sqrt{w-23}=0$$

Divide both sides by 10. Zero divided by any non-zero number gives zero.

$$\sqrt{w-23}=0$$

Square both sides of the equation.

$$w-23=0$$

Add 23 to both sides of the equation.

$$w-23-\left(-23\right)=-\left(-23\right)$$

Subtracting -23 from itself leaves 0.

$$w=-\left(-23\right)$$

Subtract -23 from 0.

$$w=23$$

Substitute 23 for w in the equation \sqrt{w+2}+\sqrt{w-23}=5.

$$\sqrt{23+2}+\sqrt{23-23}=5$$

Simplify. The value w=23 satisfies the equation.

$$5=5$$

Equation \sqrt{w+2}=-\sqrt{w-23}+5 has a unique solution.

$$w=23$$