Solve sqrt{x}+sqrt{1+x}=3+sqrt{10}

$$\sqrt{ x } + \sqrt{ 1+x } = 3+ \sqrt{ 10 }$$

Solve for x

$$x=9$$

Solution Steps

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

$$\sqrt{x}=3+\sqrt{10}-\sqrt{1+x}$$

Reorder the terms.

$$\sqrt{x}=-\sqrt{x+1}+\sqrt{10}+3$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(-\sqrt{x+1}+\sqrt{10}+3\right)^{2}$$

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

$$x=\left(-\sqrt{x+1}+\sqrt{10}+3\right)^{2}$$

Square -\sqrt{x+1}+\sqrt{10}+3.

$$x=\left(\sqrt{x+1}\right)^{2}-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+\left(\sqrt{10}\right)^{2}+6\sqrt{10}+9$$

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

$$x=x+1-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+\left(\sqrt{10}\right)^{2}+6\sqrt{10}+9$$

The square of \sqrt{10} is 10.

$$x=x+1-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+10+6\sqrt{10}+9$$

Add 1 and 10 to get 11.

$$x=x+11-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+6\sqrt{10}+9$$

Add 11 and 9 to get 20.

$$x=x+20-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+6\sqrt{10}$$

Subtract x from both sides.

$$x-x=20-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+6\sqrt{10}$$

Combine x and -x to get 0.

$$0=20-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+6\sqrt{10}$$

Swap sides so that all variable terms are on the left hand side.

$$20-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+6\sqrt{10}=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}+6\sqrt{10}=-20$$

Subtract 6\sqrt{10} from both sides.

$$-2\sqrt{10}\sqrt{x+1}-6\sqrt{x+1}=-20-6\sqrt{10}$$

Combine all terms containing x.

$$\left(-2\sqrt{10}-6\right)\sqrt{x+1}=-20-6\sqrt{10}$$

Divide both sides by -2\sqrt{10}-6.

$$\frac{\left(-2\sqrt{10}-6\right)\sqrt{x+1}}{-2\sqrt{10}-6}=\frac{-6\sqrt{10}-20}{-2\sqrt{10}-6}$$

Dividing by -2\sqrt{10}-6 undoes the multiplication by -2\sqrt{10}-6.

$$\sqrt{x+1}=\frac{-6\sqrt{10}-20}{-2\sqrt{10}-6}$$

Divide -20-6\sqrt{10} by -2\sqrt{10}-6.

$$\sqrt{x+1}=\sqrt{10}$$

Square both sides of the equation.

$$x+1=10$$

Subtract 1 from both sides of the equation.

$$x+1-1=10-1$$

Subtracting 1 from itself leaves 0.

$$x=10-1$$

Subtract 1 from 10.

$$x=9$$

Substitute 9 for x in the equation \sqrt{x}+\sqrt{1+x}=3+\sqrt{10}.

$$\sqrt{9}+\sqrt{1+9}=3+\sqrt{10}$$

Simplify. The value x=9 satisfies the equation.

$$3+10^{\frac{1}{2}}=3+10^{\frac{1}{2}}$$

Equation \sqrt{x}=-\sqrt{x+1}+\sqrt{10}+3 has a unique solution.

$$x=9$$