Solve sqrt{x+1}=sqrt{11x-8}-3sqrt{x-2}

$$\sqrt { x + 1 } = \sqrt { 11 x - 8 } - 3 \sqrt { x - 2 }$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+1}\right)^{2}=\left(\sqrt{11x-8}-3\sqrt{x-2}\right)^{2}$$

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

$$x+1=\left(\sqrt{11x-8}-3\sqrt{x-2}\right)^{2}$$

Subtract \left(\sqrt{11x-8}-3\sqrt{x-2}\right)^{2} from both sides.

$$x+1-\left(\sqrt{11x-8}-3\sqrt{x-2}\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{11x-8}-3\sqrt{x-2}\right)^{2}.

$$x+1-\left(\left(\sqrt{11x-8}\right)^{2}-6\sqrt{11x-8}\sqrt{x-2}+9\left(\sqrt{x-2}\right)^{2}\right)=0$$

Calculate \sqrt{11x-8} to the power of 2 and get 11x-8.

$$x+1-\left(11x-8-6\sqrt{11x-8}\sqrt{x-2}+9\left(\sqrt{x-2}\right)^{2}\right)=0$$

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

$$x+1-\left(11x-8-6\sqrt{11x-8}\sqrt{x-2}+9\left(x-2\right)\right)=0$$

Use the distributive property to multiply 9 by x-2.

$$x+1-\left(11x-8-6\sqrt{11x-8}\sqrt{x-2}+9x-18\right)=0$$

Combine 11x and 9x to get 20x.

$$x+1-\left(20x-8-6\sqrt{11x-8}\sqrt{x-2}-18\right)=0$$

Subtract 18 from -8 to get -26.

$$x+1-\left(20x-26-6\sqrt{11x-8}\sqrt{x-2}\right)=0$$

To find the opposite of 20x-26-6\sqrt{11x-8}\sqrt{x-2}, find the opposite of each term.

$$x+1-20x+26+6\sqrt{11x-8}\sqrt{x-2}=0$$

Combine x and -20x to get -19x.

$$-19x+1+26+6\sqrt{11x-8}\sqrt{x-2}=0$$

Add 1 and 26 to get 27.

$$-19x+27+6\sqrt{11x-8}\sqrt{x-2}=0$$

Subtract 27 from both sides. Anything subtracted from zero gives its negation.

$$-19x+6\sqrt{11x-8}\sqrt{x-2}=-27$$

Subtract -19x from both sides of the equation.

$$6\sqrt{11x-8}\sqrt{x-2}=-27+19x$$

Square both sides of the equation.

$$\left(6\sqrt{11x-8}\sqrt{x-2}\right)^{2}=\left(19x-27\right)^{2}$$

Expand \left(6\sqrt{11x-8}\sqrt{x-2}\right)^{2}.

$$6^{2}\left(\sqrt{11x-8}\right)^{2}\left(\sqrt{x-2}\right)^{2}=\left(19x-27\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$36\left(\sqrt{11x-8}\right)^{2}\left(\sqrt{x-2}\right)^{2}=\left(19x-27\right)^{2}$$

Calculate \sqrt{11x-8} to the power of 2 and get 11x-8.

$$36\left(11x-8\right)\left(\sqrt{x-2}\right)^{2}=\left(19x-27\right)^{2}$$

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

$$36\left(11x-8\right)\left(x-2\right)=\left(19x-27\right)^{2}$$

Use the distributive property to multiply 36 by 11x-8.

$$\left(396x-288\right)\left(x-2\right)=\left(19x-27\right)^{2}$$

Use the distributive property to multiply 396x-288 by x-2 and combine like terms.

$$396x^{2}-1080x+576=\left(19x-27\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(19x-27\right)^{2}.

$$396x^{2}-1080x+576=361x^{2}-1026x+729$$

Subtract 361x^{2} from both sides.

$$396x^{2}-1080x+576-361x^{2}=-1026x+729$$

Combine 396x^{2} and -361x^{2} to get 35x^{2}.

$$35x^{2}-1080x+576=-1026x+729$$

Add 1026x to both sides.

$$35x^{2}-1080x+576+1026x=729$$

Combine -1080x and 1026x to get -54x.

$$35x^{2}-54x+576=729$$

Subtract 729 from both sides.

$$35x^{2}-54x+576-729=0$$

Subtract 729 from 576 to get -153.

$$35x^{2}-54x-153=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 35x^{2}+ax+bx-153. To find a and b, set up a system to be solved.

$$a+b=-54 ab=35\left(-153\right)=-5355$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -5355.

$$1,-5355 3,-1785 5,-1071 7,-765 9,-595 15,-357 17,-315 21,-255 35,-153 45,-119 51,-105 63,-85$$

Calculate the sum for each pair.

$$1-5355=-5354 3-1785=-1782 5-1071=-1066 7-765=-758 9-595=-586 15-357=-342 17-315=-298 21-255=-234 35-153=-118 45-119=-74 51-105=-54 63-85=-22$$

The solution is the pair that gives sum -54.

$$a=-105 b=51$$

Rewrite 35x^{2}-54x-153 as \left(35x^{2}-105x\right)+\left(51x-153\right).

$$\left(35x^{2}-105x\right)+\left(51x-153\right)$$

Factor out 35x in the first and 51 in the second group.

$$35x\left(x-3\right)+51\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(35x+51\right)$$

To find equation solutions, solve x-3=0 and 35x+51=0.

$$x=3 x=-\frac{51}{35}$$

Substitute -\frac{51}{35} for x in the equation \sqrt{x+1}=\sqrt{11x-8}-3\sqrt{x-2}. The expression \sqrt{-\frac{51}{35}+1} is undefined because the radicand cannot be negative.

$$\sqrt{-\frac{51}{35}+1}=\sqrt{11\left(-\frac{51}{35}\right)-8}-3\sqrt{-\frac{51}{35}-2}$$

Substitute 3 for x in the equation \sqrt{x+1}=\sqrt{11x-8}-3\sqrt{x-2}.

$$\sqrt{3+1}=\sqrt{11\times 3-8}-3\sqrt{3-2}$$

Simplify. The value x=3 satisfies the equation.

$$2=2$$

Equation \sqrt{x+1}=\sqrt{11x-8}-3\sqrt{x-2} has a unique solution.

$$x=3$$