Question

Solve this equation with an extraneous solution. Find the real solution, extraneous solution

V44-2x = x - 10

Answer 1

solution is x = 14, extraneous solution is x = 4

Explanation:

given

`√(44-2x)` = x - 10 ( square both sides to clear the radical )

(
`√(44-2x)` )² = (x - 10)² ← expand using FOIL

44 - 2x = x² - 20x + 100 ( add 2x to both sides )

44 = x² - 18x + 100 ( subtract 44 from both sides )

0 = x² - 18x + 56 ← in standard form

0 = (x - 4)(x - 14) ← in factored form

equate each factor to zero and solve for x

x - 4 = 0 ⇒ x = 4

x - 14 = 0 ⇒ x = 14

As a check

substitute these values into the equation and if both sides are equal, then it is a solution

x = 4

left side =
`√(44-2(4))` =
`√(44-8)` =
`√(36)` = 6

right side = 4 - 10 = - 6

6 ≠ - 6

Thus x = 4 is an extraneous solution

x = 14

left side =
`√(44-2(14))` =
`√(44-28)` =
`√(16)` = 4

right side = 14 - 10 = 4

Then solution is x = 14

Answer 2

there is no extraneous solution.

Explanation:

To solve the equation V44 - 2x = x - 10, we can start by isolating the variable x on one side of the equation:

V44 - 2x = x - 10

Adding 2x to both sides:

V44 = 3x - 10

Next, we can isolate the term containing x by adding 10 to both sides:

V54 = 3x

Finally, we can solve for x by dividing both sides by 3:

x = V54/3

Now, let's evaluate the solution:

x = V54/3 ≈ 4.899

So the real solution to the equation is approximately x = 4.899.

To check if there are any extraneous solutions, we need to substitute this value back into the original equation:

V44 - 2(4.899) = 4.899 - 10

V44 - 9.798 = -5.101

At this point, we can see that the left-hand side (V44 - 9.798) is always positive since V44 is a positive quantity.

However, the right-hand side (-5.101) is negative. Since the left-hand side is always positive and the right-hand side is negative, there are no solutions that satisfy the original equation. Therefore, there is no extraneous solution.