Solve sqrt{49+39y}-y=11

$$\sqrt { 49 + 39 y } - y = 11$$

Solve for y

$$y=9$$

$$y=8$$

Solution Steps

Subtract -y from both sides of the equation.

$$\sqrt{49+39y}=11+y$$

Square both sides of the equation.

$$\left(\sqrt{49+39y}\right)^{2}=\left(11+y\right)^{2}$$

Calculate \sqrt{49+39y} to the power of 2 and get 49+39y.

$$49+39y=\left(11+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(11+y\right)^{2}.

$$49+39y=121+22y+y^{2}$$

Subtract 121 from both sides.

$$49+39y-121=22y+y^{2}$$

Subtract 121 from 49 to get -72.

$$-72+39y=22y+y^{2}$$

Subtract 22y from both sides.

$$-72+39y-22y=y^{2}$$

Combine 39y and -22y to get 17y.

$$-72+17y=y^{2}$$

Subtract y^{2} from both sides.

$$-72+17y-y^{2}=0$$

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

$$-y^{2}+17y-72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -y^{2}+ay+by-72. To find a and b, set up a system to be solved.

$$a+b=17 ab=-\left(-72\right)=72$$

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 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 17.

$$a=9 b=8$$

Rewrite -y^{2}+17y-72 as \left(-y^{2}+9y\right)+\left(8y-72\right).

$$\left(-y^{2}+9y\right)+\left(8y-72\right)$$

Factor out -y in the first and 8 in the second group.

$$-y\left(y-9\right)+8\left(y-9\right)$$

Factor out common term y-9 by using distributive property.

$$\left(y-9\right)\left(-y+8\right)$$

To find equation solutions, solve y-9=0 and -y+8=0.

$$y=9 y=8$$

Substitute 9 for y in the equation \sqrt{49+39y}-y=11.

$$\sqrt{49+39\times 9}-9=11$$

Simplify. The value y=9 satisfies the equation.

$$11=11$$

Substitute 8 for y in the equation \sqrt{49+39y}-y=11.

$$\sqrt{49+39\times 8}-8=11$$

Simplify. The value y=8 satisfies the equation.

$$11=11$$

List all solutions of \sqrt{39y+49}=y+11.

$$y=9 y=8$$