Solve 6sqrt{y+3}=14-2y

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

Solve for y

$$y=1$$

Solution Steps

Square both sides of the equation.

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

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

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

Calculate 6 to the power of 2 and get 36.

$$36\left(\sqrt{y+3}\right)^{2}=\left(14-2y\right)^{2}$$

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

$$36\left(y+3\right)=\left(14-2y\right)^{2}$$

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

$$36y+108=\left(14-2y\right)^{2}$$

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

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

Subtract 196 from both sides.

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

Subtract 196 from 108 to get -88.

$$36y-88=-56y+4y^{2}$$

Add 56y to both sides.

$$36y-88+56y=4y^{2}$$

Combine 36y and 56y to get 92y.

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

Subtract 4y^{2} from both sides.

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

Divide both sides by 4.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

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.

$$a+b=23 ab=-\left(-22\right)=22$$

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 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 -y^{2}+22y.

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

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

$$6\sqrt{22+3}=14-2\times 22$$

Simplify. The value y=22 does not satisfy the equation because the left and the right hand side have opposite signs.

$$30=-30$$

Substitute 1 for y in the equation 6\sqrt{y+3}=14-2y.

$$6\sqrt{1+3}=14-2$$

Simplify. The value y=1 satisfies the equation.

$$12=12$$

Equation 6\sqrt{y+3}=14-2y has a unique solution.

$$y=1$$