The extraneous solution to the given equation √(2p+1) + 2√p = 1 is x = 4.
What is an extraneous solution of an equation?
"An extraneous solution is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem."
Given equation is:
√(2p+1) + 2√p = 1
⇒ √(2p+1) = 1 - 2√p
⇒ (√(2p+1))² = (1 - 2√p)²
⇒ (2p + 1) = 1 - 4√p + 4p
⇒ 2p + 1 - 1 - 4p = - 4√p
⇒ -2p = - 4√p
⇒ p = 2√p
⇒ p² = 4p
⇒ p² - 4p = 0
⇒ p(p - 4) = 0
⇒ p = 0, 4
Now, putting the values of 'p' in the given equation, we get:
√(2p+1) + 2√p
= √(2 × 4+1) + 2√4
= √9 + 4
= 3 + 4 (if we take the positive sign only)
= 7
Therefore, p = 4 doesn't satisfy the given equation.
Therefore, p = 4 is the extraneous solution to the equation.
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