Solve sqrt{3y-2}+sqrt{y+3}-3=0

$$\sqrt { 3 y - 2 } + \sqrt { y + 3 } - 3 = 0$$

Solve for y

$$y=1$$

Solution Steps

Subtract \sqrt{y+3}-3 from both sides of the equation.

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

To find the opposite of \sqrt{y+3}-3, find the opposite of each term.

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

The opposite of -3 is 3.

$$\sqrt{3y-2}=-\sqrt{y+3}+3$$

Square both sides of the equation.

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

Calculate \sqrt{3y-2} to the power of 2 and get 3y-2.

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

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

$$3y-2=\left(\sqrt{y+3}\right)^{2}-6\sqrt{y+3}+9$$

Calculate \sqrt{y+3} to the power of 2 and get y+3.

$$3y-2=y+3-6\sqrt{y+3}+9$$

Add 3 and 9 to get 12.

$$3y-2=y+12-6\sqrt{y+3}$$

Subtract y+12 from both sides of the equation.

$$3y-2-\left(y+12\right)=-6\sqrt{y+3}$$

To find the opposite of y+12, find the opposite of each term.

$$3y-2-y-12=-6\sqrt{y+3}$$

Combine 3y and -y to get 2y.

$$2y-2-12=-6\sqrt{y+3}$$

Subtract 12 from -2 to get -14.

$$2y-14=-6\sqrt{y+3}$$

Square both sides of the equation.

$$\left(2y-14\right)^{2}=\left(-6\sqrt{y+3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2y-14\right)^{2}.

$$4y^{2}-56y+196=\left(-6\sqrt{y+3}\right)^{2}$$

Expand \left(-6\sqrt{y+3}\right)^{2}.

$$4y^{2}-56y+196=\left(-6\right)^{2}\left(\sqrt{y+3}\right)^{2}$$

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

$$4y^{2}-56y+196=36\left(\sqrt{y+3}\right)^{2}$$

Calculate \sqrt{y+3} to the power of 2 and get y+3.

$$4y^{2}-56y+196=36\left(y+3\right)$$

Use the distributive property to multiply 36 by y+3.

$$4y^{2}-56y+196=36y+108$$

Subtract 36y from both sides.

$$4y^{2}-56y+196-36y=108$$

Combine -56y and -36y to get -92y.

$$4y^{2}-92y+196=108$$

Subtract 108 from both sides.

$$4y^{2}-92y+196-108=0$$

Subtract 108 from 196 to get 88.

$$4y^{2}-92y+88=0$$

Divide both sides by 4.

$$y^{2}-23y+22=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by+22. To find a and b, set up a system to be solved.

$$a+b=-23 ab=1\times 22=22$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 22.

$$-1,-22 -2,-11$$

Calculate the sum for each pair.

$$-1-22=-23 -2-11=-13$$

The solution is the pair that gives sum -23.

$$a=-22 b=-1$$

Rewrite y^{2}-23y+22 as \left(y^{2}-22y\right)+\left(-y+22\right).

$$\left(y^{2}-22y\right)+\left(-y+22\right)$$

Factor out y in the first and -1 in the second group.

$$y\left(y-22\right)-\left(y-22\right)$$

Factor out common term y-22 by using distributive property.

$$\left(y-22\right)\left(y-1\right)$$

To find equation solutions, solve y-22=0 and y-1=0.

$$y=22 y=1$$

Substitute 22 for y in the equation \sqrt{3y-2}+\sqrt{y+3}-3=0.

$$\sqrt{3\times 22-2}+\sqrt{22+3}-3=0$$

Simplify. The value y=22 does not satisfy the equation.

$$10=0$$

Substitute 1 for y in the equation \sqrt{3y-2}+\sqrt{y+3}-3=0.

$$\sqrt{3\times 1-2}+\sqrt{1+3}-3=0$$

Simplify. The value y=1 satisfies the equation.

$$0=0$$

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

$$y=1$$