Solve sqrt{4x+5}-sqrt{x+1}=1

$$\sqrt { 4 x + 5 } - \sqrt { x + 1 } = 1$$

Solve for x

$$x=-\frac{5}{9}\approx -0.555555556$$

$$x=-1$$

Solution Steps

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

$$\sqrt{4x+5}=1+\sqrt{x+1}$$

Square both sides of the equation.

$$\left(\sqrt{4x+5}\right)^{2}=\left(1+\sqrt{x+1}\right)^{2}$$

Calculate \sqrt{4x+5} to the power of 2 and get 4x+5.

$$4x+5=\left(1+\sqrt{x+1}\right)^{2}$$

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

$$4x+5=1+2\sqrt{x+1}+\left(\sqrt{x+1}\right)^{2}$$

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

$$4x+5=1+2\sqrt{x+1}+x+1$$

Add 1 and 1 to get 2.

$$4x+5=2+2\sqrt{x+1}+x$$

Subtract 2+x from both sides of the equation.

$$4x+5-\left(2+x\right)=2\sqrt{x+1}$$

To find the opposite of 2+x, find the opposite of each term.

$$4x+5-2-x=2\sqrt{x+1}$$

Subtract 2 from 5 to get 3.

$$4x+3-x=2\sqrt{x+1}$$

Combine 4x and -x to get 3x.

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

Square both sides of the equation.

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

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

$$9x^{2}+18x+9=\left(2\sqrt{x+1}\right)^{2}$$

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

$$9x^{2}+18x+9=2^{2}\left(\sqrt{x+1}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$9x^{2}+18x+9=4\left(\sqrt{x+1}\right)^{2}$$

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

$$9x^{2}+18x+9=4\left(x+1\right)$$

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

$$9x^{2}+18x+9=4x+4$$

Subtract 4x from both sides.

$$9x^{2}+18x+9-4x=4$$

Combine 18x and -4x to get 14x.

$$9x^{2}+14x+9=4$$

Subtract 4 from both sides.

$$9x^{2}+14x+9-4=0$$

Subtract 4 from 9 to get 5.

$$9x^{2}+14x+5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=14 ab=9\times 5=45$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 45.

$$1,45 3,15 5,9$$

Calculate the sum for each pair.

$$1+45=46 3+15=18 5+9=14$$

The solution is the pair that gives sum 14.

$$a=5 b=9$$

Rewrite 9x^{2}+14x+5 as \left(9x^{2}+5x\right)+\left(9x+5\right).

$$\left(9x^{2}+5x\right)+\left(9x+5\right)$$

Factor out x in 9x^{2}+5x.

$$x\left(9x+5\right)+9x+5$$

Factor out common term 9x+5 by using distributive property.

$$\left(9x+5\right)\left(x+1\right)$$

To find equation solutions, solve 9x+5=0 and x+1=0.

$$x=-\frac{5}{9} x=-1$$

Substitute -\frac{5}{9} for x in the equation \sqrt{4x+5}-\sqrt{x+1}=1.

$$\sqrt{4\left(-\frac{5}{9}\right)+5}-\sqrt{-\frac{5}{9}+1}=1$$

Simplify. The value x=-\frac{5}{9} satisfies the equation.

$$1=1$$

Substitute -1 for x in the equation \sqrt{4x+5}-\sqrt{x+1}=1.

$$\sqrt{4\left(-1\right)+5}-\sqrt{-1+1}=1$$

Simplify. The value x=-1 satisfies the equation.

$$1=1$$

List all solutions of \sqrt{4x+5}=\sqrt{x+1}+1.

$$x=-\frac{5}{9} x=-1$$