Solve sqrt{2w^2-19w+31}+2=7-2w

$$\sqrt { 2 w ^ { 2 } - 19 w + 31 } + 2 = 7 - 2 w$$

Solve for w

$$w=2$$

$$w = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Solution Steps

Subtract 2 from both sides of the equation.

$$\sqrt{2w^{2}-19w+31}=7-2w-2$$

Subtract 2 from 7 to get 5.

$$\sqrt{2w^{2}-19w+31}=5-2w$$

Square both sides of the equation.

$$\left(\sqrt{2w^{2}-19w+31}\right)^{2}=\left(5-2w\right)^{2}$$

Calculate \sqrt{2w^{2}-19w+31} to the power of 2 and get 2w^{2}-19w+31.

$$2w^{2}-19w+31=\left(5-2w\right)^{2}$$

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

$$2w^{2}-19w+31=25-20w+4w^{2}$$

Subtract 25 from both sides.

$$2w^{2}-19w+31-25=-20w+4w^{2}$$

Subtract 25 from 31 to get 6.

$$2w^{2}-19w+6=-20w+4w^{2}$$

Add 20w to both sides.

$$2w^{2}-19w+6+20w=4w^{2}$$

Combine -19w and 20w to get w.

$$2w^{2}+w+6=4w^{2}$$

Subtract 4w^{2} from both sides.

$$2w^{2}+w+6-4w^{2}=0$$

Combine 2w^{2} and -4w^{2} to get -2w^{2}.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2w^{2}+aw+bw+6. To find a and b, set up a system to be solved.

$$a+b=1 ab=-2\times 6=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 1.

$$a=4 b=-3$$

Rewrite -2w^{2}+w+6 as \left(-2w^{2}+4w\right)+\left(-3w+6\right).

$$\left(-2w^{2}+4w\right)+\left(-3w+6\right)$$

Factor out 2w in the first and 3 in the second group.

$$2w\left(-w+2\right)+3\left(-w+2\right)$$

Factor out common term -w+2 by using distributive property.

$$\left(-w+2\right)\left(2w+3\right)$$

To find equation solutions, solve -w+2=0 and 2w+3=0.

$$w=2 w=-\frac{3}{2}$$

Substitute 2 for w in the equation \sqrt{2w^{2}-19w+31}+2=7-2w.

$$\sqrt{2\times 2^{2}-19\times 2+31}+2=7-2\times 2$$

Simplify. The value w=2 satisfies the equation.

$$3=3$$

Substitute -\frac{3}{2} for w in the equation \sqrt{2w^{2}-19w+31}+2=7-2w.

$$\sqrt{2\left(-\frac{3}{2}\right)^{2}-19\left(-\frac{3}{2}\right)+31}+2=7-2\left(-\frac{3}{2}\right)$$

Simplify. The value w=-\frac{3}{2} satisfies the equation.

$$10=10$$

List all solutions of \sqrt{2w^{2}-19w+31}=5-2w.

$$w=2 w=-\frac{3}{2}$$