Solve sqrt{3w+37}-3=w

$$\sqrt { 3 w + 37 } - 3 = w$$

Solve for w

$$w=4$$

Solution Steps

Subtract -3 from both sides of the equation.

$$\sqrt{3w+37}=w+3$$

Square both sides of the equation.

$$\left(\sqrt{3w+37}\right)^{2}=\left(w+3\right)^{2}$$

Calculate \sqrt{3w+37} to the power of 2 and get 3w+37.

$$3w+37=\left(w+3\right)^{2}$$

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

$$3w+37=w^{2}+6w+9$$

Subtract w^{2} from both sides.

$$3w+37-w^{2}=6w+9$$

Subtract 6w from both sides.

$$3w+37-w^{2}-6w=9$$

Combine 3w and -6w to get -3w.

$$-3w+37-w^{2}=9$$

Subtract 9 from both sides.

$$-3w+37-w^{2}-9=0$$

Subtract 9 from 37 to get 28.

$$-3w+28-w^{2}=0$$

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

$$-w^{2}-3w+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -w^{2}+aw+bw+28. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-28=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Calculate the sum for each pair.

$$1-28=-27 2-14=-12 4-7=-3$$

The solution is the pair that gives sum -3.

$$a=4 b=-7$$

Rewrite -w^{2}-3w+28 as \left(-w^{2}+4w\right)+\left(-7w+28\right).

$$\left(-w^{2}+4w\right)+\left(-7w+28\right)$$

Factor out w in the first and 7 in the second group.

$$w\left(-w+4\right)+7\left(-w+4\right)$$

Factor out common term -w+4 by using distributive property.

$$\left(-w+4\right)\left(w+7\right)$$

To find equation solutions, solve -w+4=0 and w+7=0.

$$w=4 w=-7$$

Substitute 4 for w in the equation \sqrt{3w+37}-3=w.

$$\sqrt{3\times 4+37}-3=4$$

Simplify. The value w=4 satisfies the equation.

$$4=4$$

Substitute -7 for w in the equation \sqrt{3w+37}-3=w.

$$\sqrt{3\left(-7\right)+37}-3=-7$$

Simplify. The value w=-7 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-7$$

Equation \sqrt{3w+37}=w+3 has a unique solution.

$$w=4$$