Solve sqrt{x-4}+sqrt{x+5}=sqrt{11}

$$\sqrt { x - 4 } + \sqrt { x + 5 } = \sqrt { 11 }$$

Solve for x

$$x = \frac{45}{11} = 4\frac{1}{11} \approx 4.090909091$$

Solution Steps

Subtract \sqrt{x+5} from both sides of the equation.

$$\sqrt{x-4}=\sqrt{11}-\sqrt{x+5}$$

Square both sides of the equation.

$$\left(\sqrt{x-4}\right)^{2}=\left(\sqrt{11}-\sqrt{x+5}\right)^{2}$$

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

$$x-4=\left(\sqrt{11}-\sqrt{x+5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{11}-\sqrt{x+5}\right)^{2}.

$$x-4=\left(\sqrt{11}\right)^{2}-2\sqrt{11}\sqrt{x+5}+\left(\sqrt{x+5}\right)^{2}$$

The square of \sqrt{11} is 11.

$$x-4=11-2\sqrt{11}\sqrt{x+5}+\left(\sqrt{x+5}\right)^{2}$$

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

$$x-4=11-2\sqrt{11}\sqrt{x+5}+x+5$$

Add 11 and 5 to get 16.

$$x-4=16-2\sqrt{11}\sqrt{x+5}+x$$

Add 2\sqrt{11}\sqrt{x+5} to both sides.

$$x-4+2\sqrt{11}\sqrt{x+5}=16+x$$

Subtract x from both sides.

$$x-4+2\sqrt{11}\sqrt{x+5}-x=16$$

Combine x and -x to get 0.

$$-4+2\sqrt{11}\sqrt{x+5}=16$$

Add 4 to both sides.

$$2\sqrt{11}\sqrt{x+5}=16+4$$

Add 16 and 4 to get 20.

$$2\sqrt{11}\sqrt{x+5}=20$$

Divide both sides by 2\sqrt{11}.

$$\frac{2\sqrt{11}\sqrt{x+5}}{2\sqrt{11}}=\frac{20}{2\sqrt{11}}$$

Dividing by 2\sqrt{11} undoes the multiplication by 2\sqrt{11}.

$$\sqrt{x+5}=\frac{20}{2\sqrt{11}}$$

Divide 20 by 2\sqrt{11}.

$$\sqrt{x+5}=\frac{10\sqrt{11}}{11}$$

Square both sides of the equation.

$$x+5=\frac{100}{11}$$

Subtract 5 from both sides of the equation.

$$x+5-5=\frac{100}{11}-5$$

Subtracting 5 from itself leaves 0.

$$x=\frac{100}{11}-5$$

Subtract 5 from \frac{100}{11}.

$$x=\frac{45}{11}$$

Substitute \frac{45}{11} for x in the equation \sqrt{x-4}+\sqrt{x+5}=\sqrt{11}.

$$\sqrt{\frac{45}{11}-4}+\sqrt{\frac{45}{11}+5}=\sqrt{11}$$

Simplify. The value x=\frac{45}{11} satisfies the equation.

$$11^{\frac{1}{2}}=11^{\frac{1}{2}}$$

Equation \sqrt{x-4}=-\sqrt{x+5}+\sqrt{11} has a unique solution.

$$x=\frac{45}{11}$$