How do you solve #w = sqrt[7w] # and find any extraneous solutions?

Question

Avery Alexander · Accepted Answer

#w=0 and w=7#

However we have a square root so for the solution #in RRcolor(white)("d"); w>=0#

Given: #w=sqrt(7w)#

Square both sides

#w^2=7w#

Subtract #7w# from both sides

#w^2-7w+0=0#

And we have our quadratic!

compare to #aw^2+bw+c=0 color(white)("ddd") ->color(white)("ddd") w=(-b+-sqrt(b^2-4ac))/(2a)#

#w=(7+-sqrt((-7)^2-4(1)(0)))/(2(1))#

#w=7/2+-7/2#

#w=0 and w=7#

Eva Abramson · Answer

undefined To solve the equation w = sqrt[7w] and find any extraneous solutions, we can follow these steps:

1. Square both sides of the equation to eliminate the square root: w^2 = 7w.
2. Rearrange the equation to bring all terms to one side: w^2 - 7w = 0.
3. Factor out w: w(w - 7) = 0.
4. Set each factor equal to zero and solve for w: w = 0 or w - 7 = 0.
5. Solve for w in the second equation: w = 7.
6. The solutions to the original equation are w = 0 and w = 7.
7. To check for extraneous solutions, substitute each solution back into the original equation.
   - For w = 0: sqrt[7(0)] = sqrt[0] = 0. The equation is satisfied.
   - For w = 7: sqrt[7(7)] = sqrt[49] = 7. The equation is satisfied.
8. There are no extraneous solutions in this case. The solutions are w = 0 and w = 7.