Solve sqrt{10+2x}=x+1

$$\sqrt { 10 + 2 x } = x + 1$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{10+2x} to the power of 2 and get 10+2x.

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

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

$$10+2x=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$10+2x-x^{2}=2x+1$$

Subtract 2x from both sides.

$$10+2x-x^{2}-2x=1$$

Combine 2x and -2x to get 0.

$$10-x^{2}=1$$

Subtract 10 from both sides.

$$-x^{2}=1-10$$

Subtract 10 from 1 to get -9.

$$-x^{2}=-9$$

Divide both sides by -1.

$$x^{2}=\frac{-9}{-1}$$

Fraction \frac{-9}{-1} can be simplified to 9 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=9$$

Take the square root of both sides of the equation.

$$x=3 x=-3$$

Substitute 3 for x in the equation \sqrt{10+2x}=x+1.

$$\sqrt{10+2\times 3}=3+1$$

Simplify. The value x=3 satisfies the equation.

$$4=4$$

Substitute -3 for x in the equation \sqrt{10+2x}=x+1.

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

Simplify. The value x=-3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{2x+10}=x+1 has a unique solution.

$$x=3$$