SOLUTION: Find all solutions of the equation |x^2 - 30x - 1| = 30 + |6x - 2| + |x^2 - 20x - 4|.

Algebra.Com
Question 1209411: Find all solutions of the equation |x^2 - 30x - 1| = 30 + |6x - 2| + |x^2 - 20x - 4|.
Found 3 solutions by Edwin McCravy, KMST, ikleyn:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!



Get 0 on the right:



Graph the left side in your TI-84 calculator. 

The "abs" feature is found this way: MATH NUM 1:abs(

Put it in Y1 like this:

abs(x^2 - 30x - 1) - 30 - abs(6x - 2) - abs(x^2 - 20x - 4)

Set the window Xmin=-10, Xmax=25, Ymin=-40, Ymax=60

Your graph will look something close to this:



Use the zero feature of the calculator, and you'll find three solutions

The "zero" feature is found 2ND TRACE 2:zero

x = -7.25,   x = 7.75,  and x = 21.464225

Edwin
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
Before graphing calculators, it would have been a more complicated

The right hand is greater than 30 except for all values of
for all of real values because
for all real numbers,
with for , and otherwise.
for
and .
and for
for such that

For the left hand side, , has as solutions
For values of and ,
and .
For
Those limiting values are approximately and .

For , , while and
and the equation turns into


-->

For is decreasing from at to a minimum 0f at and then increasing to at and is less than the value of
=
For , , , and ,
so and
The equation turns into


-->

For while
and , so
turning the equation into

-->
does not comply with
is a solution between and

For , , and ,
turning the equation into

--> is not a solution complying with
Answer by ikleyn(52858)   (Show Source): You can put this solution on YOUR website!
.

For problems like this one, there is a standard methodology/strategy for solving.

You should subdivide the number line by sections (= intervals).
The division point are the points where the participating functions under absolute values
(parts of the equation) change their signs.

Then for each interval, from minus infinity to infinity, you write an equation
using each function with its native sign as it should be, dictated by the absolute
value rule at this interval.

Then you solve the updated equation and check if the solution does belong to this particular
interval under the consideration.

If the particular solution does belong to this particular interval, then the found solution
is the solution to the original equation.
If it does not belong, then the found solution is not a solution to the original equation.

As you complete going from the left to the right , you complete the solution.

The rest is just a technique and the arithmetic.


I think, if a student does understand it and demonstrates his/her understanding, it should be just
enough for the teacher to accept the assignment.


As the problem is twisted in this post, it is twisted TOO much and teaches nothing.


A teacher who assigns such tasks, should be checked for adequacy.



RELATED QUESTIONS

Find all solutions of the equation|x^2-20x-1|=20. Please explain in... (answered by ikleyn)
Find all the solutions of the equation {{{(x^2 -7x+11)^(x^2-11x+30)... (answered by Alan3354)
find all real solutions of the equation.... (answered by jim_thompson5910)
Find all solutions to the equation (x - 1)^2 =... (answered by t0hierry)
Find all solutions to the equation $(x - 1)^2 =... (answered by ikleyn)
Find all real solutions of the equation (x)square/x-1 + 1/x+2 = 5/4 (answered by Nate)
1/x-1 + 1/x+2 = 5/4 Find all real solutions of the... (answered by Nate)
Find all real soultions. x^2+30x + 200 = 0 x^2 + 3x + 1 = 0 X^2- 6x + 1 = 0... (answered by jim_thompson5910)
Find all real solutions of the equation.... (answered by Boreal)