Solve sqrt{x+5}+sqrt{x+21}=sqrt{x+60}

$$\sqrt{ x+5 } + \sqrt{ x+21 } = \sqrt{ x+60 }$$

Solve for x

$$x=4$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+5}+\sqrt{x+21}\right)^{2}=\left(\sqrt{x+60}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x+5}+\sqrt{x+21}\right)^{2}.

$$\left(\sqrt{x+5}\right)^{2}+2\sqrt{x+5}\sqrt{x+21}+\left(\sqrt{x+21}\right)^{2}=\left(\sqrt{x+60}\right)^{2}$$

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

$$x+5+2\sqrt{x+5}\sqrt{x+21}+\left(\sqrt{x+21}\right)^{2}=\left(\sqrt{x+60}\right)^{2}$$

Calculate \sqrt{x+21} to the power of 2 and get x+21.

$$x+5+2\sqrt{x+5}\sqrt{x+21}+x+21=\left(\sqrt{x+60}\right)^{2}$$

Combine x and x to get 2x.

$$2x+5+2\sqrt{x+5}\sqrt{x+21}+21=\left(\sqrt{x+60}\right)^{2}$$

Add 5 and 21 to get 26.

$$2x+26+2\sqrt{x+5}\sqrt{x+21}=\left(\sqrt{x+60}\right)^{2}$$

Calculate \sqrt{x+60} to the power of 2 and get x+60.

$$2x+26+2\sqrt{x+5}\sqrt{x+21}=x+60$$

Subtract 2x+26 from both sides of the equation.

$$2\sqrt{x+5}\sqrt{x+21}=x+60-\left(2x+26\right)$$

To find the opposite of 2x+26, find the opposite of each term.

$$2\sqrt{x+5}\sqrt{x+21}=x+60-2x-26$$

Combine x and -2x to get -x.

$$2\sqrt{x+5}\sqrt{x+21}=-x+60-26$$

Subtract 26 from 60 to get 34.

$$2\sqrt{x+5}\sqrt{x+21}=-x+34$$

Square both sides of the equation.

$$\left(2\sqrt{x+5}\sqrt{x+21}\right)^{2}=\left(-x+34\right)^{2}$$

Expand \left(2\sqrt{x+5}\sqrt{x+21}\right)^{2}.

$$2^{2}\left(\sqrt{x+5}\right)^{2}\left(\sqrt{x+21}\right)^{2}=\left(-x+34\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x+5}\right)^{2}\left(\sqrt{x+21}\right)^{2}=\left(-x+34\right)^{2}$$

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

$$4\left(x+5\right)\left(\sqrt{x+21}\right)^{2}=\left(-x+34\right)^{2}$$

Calculate \sqrt{x+21} to the power of 2 and get x+21.

$$4\left(x+5\right)\left(x+21\right)=\left(-x+34\right)^{2}$$

Use the distributive property to multiply 4 by x+5.

$$\left(4x+20\right)\left(x+21\right)=\left(-x+34\right)^{2}$$

Apply the distributive property by multiplying each term of 4x+20 by each term of x+21.

$$4x^{2}+84x+20x+420=\left(-x+34\right)^{2}$$

Combine 84x and 20x to get 104x.

$$4x^{2}+104x+420=\left(-x+34\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-x+34\right)^{2}.

$$4x^{2}+104x+420=x^{2}-68x+1156$$

Subtract x^{2} from both sides.

$$4x^{2}+104x+420-x^{2}=-68x+1156$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+104x+420=-68x+1156$$

Add 68x to both sides.

$$3x^{2}+104x+420+68x=1156$$

Combine 104x and 68x to get 172x.

$$3x^{2}+172x+420=1156$$

Subtract 1156 from both sides.

$$3x^{2}+172x+420-1156=0$$

Subtract 1156 from 420 to get -736.

$$3x^{2}+172x-736=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx-736. To find a and b, set up a system to be solved.

$$a+b=172 ab=3\left(-736\right)=-2208$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -2208.

$$-1,2208 -2,1104 -3,736 -4,552 -6,368 -8,276 -12,184 -16,138 -23,96 -24,92 -32,69 -46,48$$

Calculate the sum for each pair.

$$-1+2208=2207 -2+1104=1102 -3+736=733 -4+552=548 -6+368=362 -8+276=268 -12+184=172 -16+138=122 -23+96=73 -24+92=68 -32+69=37 -46+48=2$$

The solution is the pair that gives sum 172.

$$a=-12 b=184$$

Rewrite 3x^{2}+172x-736 as \left(3x^{2}-12x\right)+\left(184x-736\right).

$$\left(3x^{2}-12x\right)+\left(184x-736\right)$$

Factor out 3x in the first and 184 in the second group.

$$3x\left(x-4\right)+184\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(3x+184\right)$$

To find equation solutions, solve x-4=0 and 3x+184=0.

$$x=4 x=-\frac{184}{3}$$

Substitute -\frac{184}{3} for x in the equation \sqrt{x+5}+\sqrt{x+21}=\sqrt{x+60}. The expression \sqrt{-\frac{184}{3}+5} is undefined because the radicand cannot be negative.

$$\sqrt{-\frac{184}{3}+5}+\sqrt{-\frac{184}{3}+21}=\sqrt{-\frac{184}{3}+60}$$

Substitute 4 for x in the equation \sqrt{x+5}+\sqrt{x+21}=\sqrt{x+60}.

$$\sqrt{4+5}+\sqrt{4+21}=\sqrt{4+60}$$

Simplify. The value x=4 satisfies the equation.

$$8=8$$

Equation \sqrt{x+5}+\sqrt{x+21}=\sqrt{x+60} has a unique solution.

$$x=4$$