Solve sqrt{-2w+43}=w-4

$$\sqrt { - 2 w + 43 } = w - 4$$

Solve for w

$$w=9$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{-2w+43}\right)^{2}=\left(w-4\right)^{2}$$

Calculate \sqrt{-2w+43} to the power of 2 and get -2w+43.

$$-2w+43=\left(w-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(w-4\right)^{2}.

$$-2w+43=w^{2}-8w+16$$

Subtract w^{2} from both sides.

$$-2w+43-w^{2}=-8w+16$$

Add 8w to both sides.

$$-2w+43-w^{2}+8w=16$$

Combine -2w and 8w to get 6w.

$$6w+43-w^{2}=16$$

Subtract 16 from both sides.

$$6w+43-w^{2}-16=0$$

Subtract 16 from 43 to get 27.

$$6w+27-w^{2}=0$$

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

$$-w^{2}+6w+27=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -w^{2}+aw+bw+27. To find a and b, set up a system to be solved.

$$a+b=6 ab=-27=-27$$

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 -27.

$$-1,27 -3,9$$

Calculate the sum for each pair.

$$-1+27=26 -3+9=6$$

The solution is the pair that gives sum 6.

$$a=9 b=-3$$

Rewrite -w^{2}+6w+27 as \left(-w^{2}+9w\right)+\left(-3w+27\right).

$$\left(-w^{2}+9w\right)+\left(-3w+27\right)$$

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

$$-w\left(w-9\right)-3\left(w-9\right)$$

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

$$\left(w-9\right)\left(-w-3\right)$$

To find equation solutions, solve w-9=0 and -w-3=0.

$$w=9 w=-3$$

Substitute 9 for w in the equation \sqrt{-2w+43}=w-4.

$$\sqrt{-2\times 9+43}=9-4$$

Simplify. The value w=9 satisfies the equation.

$$5=5$$

Substitute -3 for w in the equation \sqrt{-2w+43}=w-4.

$$\sqrt{-2\left(-3\right)+43}=-3-4$$

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

$$7=-7$$

Equation \sqrt{43-2w}=w-4 has a unique solution.

$$w=9$$