Solve 3w-7=sqrt{8w-7}

$$3 w - 7 = \sqrt { 8 w - 7 }$$

Solve for w

$$w=4$$

Solution Steps

Square both sides of the equation.

$$\left(3w-7\right)^{2}=\left(\sqrt{8w-7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3w-7\right)^{2}.

$$9w^{2}-42w+49=\left(\sqrt{8w-7}\right)^{2}$$

Calculate \sqrt{8w-7} to the power of 2 and get 8w-7.

$$9w^{2}-42w+49=8w-7$$

Subtract 8w from both sides.

$$9w^{2}-42w+49-8w=-7$$

Combine -42w and -8w to get -50w.

$$9w^{2}-50w+49=-7$$

Add 7 to both sides.

$$9w^{2}-50w+49+7=0$$

Add 49 and 7 to get 56.

$$9w^{2}-50w+56=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9w^{2}+aw+bw+56. To find a and b, set up a system to be solved.

$$a+b=-50 ab=9\times 56=504$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 504.

$$-1,-504 -2,-252 -3,-168 -4,-126 -6,-84 -7,-72 -8,-63 -9,-56 -12,-42 -14,-36 -18,-28 -21,-24$$

Calculate the sum for each pair.

$$-1-504=-505 -2-252=-254 -3-168=-171 -4-126=-130 -6-84=-90 -7-72=-79 -8-63=-71 -9-56=-65 -12-42=-54 -14-36=-50 -18-28=-46 -21-24=-45$$

The solution is the pair that gives sum -50.

$$a=-36 b=-14$$

Rewrite 9w^{2}-50w+56 as \left(9w^{2}-36w\right)+\left(-14w+56\right).

$$\left(9w^{2}-36w\right)+\left(-14w+56\right)$$

Factor out 9w in the first and -14 in the second group.

$$9w\left(w-4\right)-14\left(w-4\right)$$

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

$$\left(w-4\right)\left(9w-14\right)$$

To find equation solutions, solve w-4=0 and 9w-14=0.

$$w=4 w=\frac{14}{9}$$

Substitute 4 for w in the equation 3w-7=\sqrt{8w-7}.

$$3\times 4-7=\sqrt{8\times 4-7}$$

Simplify. The value w=4 satisfies the equation.

$$5=5$$

Substitute \frac{14}{9} for w in the equation 3w-7=\sqrt{8w-7}.

$$3\times \frac{14}{9}-7=\sqrt{8\times \frac{14}{9}-7}$$

Simplify. The value w=\frac{14}{9} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{7}{3}=\frac{7}{3}$$

Equation 3w-7=\sqrt{8w-7} has a unique solution.

$$w=4$$