Show each step involved in solving the following equation. Tell if any of them are extraneous:
sqrt(3x + 4) + x = 8
dumb question, but what does extraneous mean?
2006-11-27 14:10:41 · 4 answers · asked by GMEN 1 in Science & Mathematics ➔ Mathematics
sqrt(3x + 4) + x = 8
sqrt(3x + 4) = 8 - x
3x + 4 = 64 - 16x + x^2
0 = x^2 - 19x + 60
(x - 15)(x - 4) = 0
x = 4 or 15
Substitute x=4 into the original equation and it works.
Substitute x=15 into the original equation and it doesn't. It is extraneous.
2006-11-28 13:54:22 · answer #1 · answered by Kemmy 6 · 2⤊ 0⤋
sqrt(3x+4) + x = 8
Subtract x from each side
sqrt(3x+4) = 8-x
Square both sides
3x + 4 = (8-x)^2
3x + 4 = 64 - 16x + x^2
Move everything to one side
x^2 - 19x + 60 = 0
Factor. Ask yourself "what factors of 60 add up to -19?"
-15 times -4 is 60 and -15-4 = -19
(x-15)(x-4) = 0
x-15 = 0 or x-4 = 0
x = 15 or x = 4
Now you must always check your solution when you have an equation with square roots. Sometimes your "solution" may not actually be a solution.
Check: x = 15
sqrt(3*15 +4) + 15 = 8
sqrt(49) + 15 = 8
49 + 15 = 8
64 = 8 Obviously this is not true. x=15 is an extraneous solution.
Check: x = 4
sqrt(3*4 +4) + 4 = 8
sqrt(16) + 4 = 8
4 + 4 = 8
8 = 8 This is true.
Answer: x = 4
2006-11-27 23:18:25 · answer #2 · answered by MsMath 7 · 0⤊ 3⤋
To solve questions like this, you have to square both sides of the equation.
It turns out that, when you do so, you actually change the equation a little.
For instance consider the following two equations:
x = sqrt(-2x + 3)
x^2 = -2x + 3
If you square both sides of the first equation, you get the second equation.
However, both -3 and 1 are solutions to the second equation.
-3 is NOT a solution to the first equation, because when you plug it back in, you'd get -3 on the left and 3 on the right.
So -3 is called an "extraneous root", because in the process of trying to solve the original equation, it became a root...but since it doesn't work when you plug it back in the original equation, you have to throw it away.
Make sense?
So, since polarbear solved it for you, try plugging both answers back in the original equation. One of them (15) doesn't work, right? That's the extraneous one.
2006-11-27 22:19:25 · answer #3 · answered by Jim Burnell 6 · 3⤊ 1⤋
It means extra, or unnecessary. If you could have gotten the answer without going through that step, it was extraneous.
2006-11-27 22:13:19 · answer #4 · answered by Amy F 5 · 1⤊ 0⤋