Solve sqrt{46-6w}=w-5

$$\sqrt { 46 - 6 w } = w - 5$$

Solve for w

$$w=7$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{46-6w}\right)^{2}=\left(w-5\right)^{2}$$

Calculate \sqrt{46-6w} to the power of 2 and get 46-6w.

$$46-6w=\left(w-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(w-5\right)^{2}.

$$46-6w=w^{2}-10w+25$$

Subtract w^{2} from both sides.

$$46-6w-w^{2}=-10w+25$$

Add 10w to both sides.

$$46-6w-w^{2}+10w=25$$

Combine -6w and 10w to get 4w.

$$46+4w-w^{2}=25$$

Subtract 25 from both sides.

$$46+4w-w^{2}-25=0$$

Subtract 25 from 46 to get 21.

$$21+4w-w^{2}=0$$

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

$$-w^{2}+4w+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -w^{2}+aw+bw+21. To find a and b, set up a system to be solved.

$$a+b=4 ab=-21=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 4.

$$a=7 b=-3$$

Rewrite -w^{2}+4w+21 as \left(-w^{2}+7w\right)+\left(-3w+21\right).

$$\left(-w^{2}+7w\right)+\left(-3w+21\right)$$

Factor out -w in the first and -3 in the second group.

$$-w\left(w-7\right)-3\left(w-7\right)$$

Factor out common term w-7 by using distributive property.

$$\left(w-7\right)\left(-w-3\right)$$

To find equation solutions, solve w-7=0 and -w-3=0.

$$w=7 w=-3$$

Substitute 7 for w in the equation \sqrt{46-6w}=w-5.

$$\sqrt{46-6\times 7}=7-5$$

Simplify. The value w=7 satisfies the equation.

$$2=2$$

Substitute -3 for w in the equation \sqrt{46-6w}=w-5.

$$\sqrt{46-6\left(-3\right)}=-3-5$$

Simplify. The value w=-3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$8=-8$$

Equation \sqrt{46-6w}=w-5 has a unique solution.

$$w=7$$