Solve sqrt{166+39y}-y=14

$$\sqrt { 166 + 39 y } - y = 14$$

Solve for y

$$y=6$$

$$y=5$$

Solution Steps

Subtract -y from both sides of the equation.

$$\sqrt{166+39y}=14+y$$

Square both sides of the equation.

$$\left(\sqrt{166+39y}\right)^{2}=\left(14+y\right)^{2}$$

Calculate \sqrt{166+39y} to the power of 2 and get 166+39y.

$$166+39y=\left(14+y\right)^{2}$$

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

$$166+39y=196+28y+y^{2}$$

Subtract 196 from both sides.

$$166+39y-196=28y+y^{2}$$

Subtract 196 from 166 to get -30.

$$-30+39y=28y+y^{2}$$

Subtract 28y from both sides.

$$-30+39y-28y=y^{2}$$

Combine 39y and -28y to get 11y.

$$-30+11y=y^{2}$$

Subtract y^{2} from both sides.

$$-30+11y-y^{2}=0$$

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

$$-y^{2}+11y-30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -y^{2}+ay+by-30. To find a and b, set up a system to be solved.

$$a+b=11 ab=-\left(-30\right)=30$$

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 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 11.

$$a=6 b=5$$

Rewrite -y^{2}+11y-30 as \left(-y^{2}+6y\right)+\left(5y-30\right).

$$\left(-y^{2}+6y\right)+\left(5y-30\right)$$

Factor out -y in the first and 5 in the second group.

$$-y\left(y-6\right)+5\left(y-6\right)$$

Factor out common term y-6 by using distributive property.

$$\left(y-6\right)\left(-y+5\right)$$

To find equation solutions, solve y-6=0 and -y+5=0.

$$y=6 y=5$$

Substitute 6 for y in the equation \sqrt{166+39y}-y=14.

$$\sqrt{166+39\times 6}-6=14$$

Simplify. The value y=6 satisfies the equation.

$$14=14$$

Substitute 5 for y in the equation \sqrt{166+39y}-y=14.

$$\sqrt{166+39\times 5}-5=14$$

Simplify. The value y=5 satisfies the equation.

$$14=14$$

List all solutions of \sqrt{39y+166}=y+14.

$$y=6 y=5$$