Solve 2*sqrt{x+1}+sqrt{x+2}=3

$$2 \cdot \sqrt { x + 1 } + \sqrt { x + 2 } = 3$$

Solve for x

$$x=\frac{13-8\sqrt{3}}{3}\approx -0.28546882$$

Solution Steps

Subtract \sqrt{x+2} from both sides of the equation.

$$2\sqrt{x+1}=3-\sqrt{x+2}$$

Square both sides of the equation.

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

Expand \left(2\sqrt{x+1}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x+1}\right)^{2}=\left(3-\sqrt{x+2}\right)^{2}$$

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

$$4\left(x+1\right)=\left(3-\sqrt{x+2}\right)^{2}$$

Use the distributive property to multiply 4 by x+1.

$$4x+4=\left(3-\sqrt{x+2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{x+2}\right)^{2}.

$$4x+4=9-6\sqrt{x+2}+\left(\sqrt{x+2}\right)^{2}$$

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

$$4x+4=9-6\sqrt{x+2}+x+2$$

Add 9 and 2 to get 11.

$$4x+4=11-6\sqrt{x+2}+x$$

Subtract 11+x from both sides of the equation.

$$4x+4-\left(11+x\right)=-6\sqrt{x+2}$$

To find the opposite of 11+x, find the opposite of each term.

$$4x+4-11-x=-6\sqrt{x+2}$$

Subtract 11 from 4 to get -7.

$$4x-7-x=-6\sqrt{x+2}$$

Combine 4x and -x to get 3x.

$$3x-7=-6\sqrt{x+2}$$

Square both sides of the equation.

$$\left(3x-7\right)^{2}=\left(-6\sqrt{x+2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-7\right)^{2}.

$$9x^{2}-42x+49=\left(-6\sqrt{x+2}\right)^{2}$$

Expand \left(-6\sqrt{x+2}\right)^{2}.

$$9x^{2}-42x+49=\left(-6\right)^{2}\left(\sqrt{x+2}\right)^{2}$$

Calculate -6 to the power of 2 and get 36.

$$9x^{2}-42x+49=36\left(\sqrt{x+2}\right)^{2}$$

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

$$9x^{2}-42x+49=36\left(x+2\right)$$

Use the distributive property to multiply 36 by x+2.

$$9x^{2}-42x+49=36x+72$$

Subtract 36x from both sides.

$$9x^{2}-42x+49-36x=72$$

Combine -42x and -36x to get -78x.

$$9x^{2}-78x+49=72$$

Subtract 72 from both sides.

$$9x^{2}-78x+49-72=0$$

Subtract 72 from 49 to get -23.

$$9x^{2}-78x-23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -78 for b, and -23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-78\right)±\sqrt{\left(-78\right)^{2}-4\times 9\left(-23\right)}}{2\times 9}$$

Square -78.

$$x=\frac{-\left(-78\right)±\sqrt{6084-4\times 9\left(-23\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-78\right)±\sqrt{6084-36\left(-23\right)}}{2\times 9}$$

Multiply -36 times -23.

$$x=\frac{-\left(-78\right)±\sqrt{6084+828}}{2\times 9}$$

Add 6084 to 828.

$$x=\frac{-\left(-78\right)±\sqrt{6912}}{2\times 9}$$

Take the square root of 6912.

$$x=\frac{-\left(-78\right)±48\sqrt{3}}{2\times 9}$$

The opposite of -78 is 78.

$$x=\frac{78±48\sqrt{3}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{78±48\sqrt{3}}{18}$$

Now solve the equation x=\frac{78±48\sqrt{3}}{18} when ± is plus. Add 78 to 48\sqrt{3}.

$$x=\frac{48\sqrt{3}+78}{18}$$

Divide 78+48\sqrt{3} by 18.

$$x=\frac{8\sqrt{3}+13}{3}$$

Now solve the equation x=\frac{78±48\sqrt{3}}{18} when ± is minus. Subtract 48\sqrt{3} from 78.

$$x=\frac{78-48\sqrt{3}}{18}$$

Divide 78-48\sqrt{3} by 18.

$$x=\frac{13-8\sqrt{3}}{3}$$

The equation is now solved.

$$x=\frac{8\sqrt{3}+13}{3} x=\frac{13-8\sqrt{3}}{3}$$

Substitute \frac{8\sqrt{3}+13}{3} for x in the equation 2\sqrt{x+1}+\sqrt{x+2}=3.

$$2\sqrt{\frac{8\sqrt{3}+13}{3}+1}+\sqrt{\frac{8\sqrt{3}+13}{3}+2}=3$$

Simplify. The value x=\frac{8\sqrt{3}+13}{3} does not satisfy the equation.

$$5+\frac{8}{3}\times 3^{\frac{1}{2}}=3$$

Substitute \frac{13-8\sqrt{3}}{3} for x in the equation 2\sqrt{x+1}+\sqrt{x+2}=3.

$$2\sqrt{\frac{13-8\sqrt{3}}{3}+1}+\sqrt{\frac{13-8\sqrt{3}}{3}+2}=3$$

Simplify. The value x=\frac{13-8\sqrt{3}}{3} satisfies the equation.

$$3=3$$

Equation 2\sqrt{x+1}=-\sqrt{x+2}+3 has a unique solution.

$$x=\frac{13-8\sqrt{3}}{3}$$