Final answer:
To solve the equation, isolate the variable p by eliminating the square roots. The extraneous solution is p = -2.
Step-by-step explanation:
To solve the equation √2p+1+2√√p=12, we need to isolate the variable p. Begin by subtracting 2√√p from both sides of the equation. This gives us √2p+1 = 12 - 2√√p. Next, square both sides of the equation to eliminate the square root. This gives us 2p+1 = (12 - 2√√p)². Simplifying further, we have 2p+1 = 144 - 48√√p + 4p. Rearrange the equation to isolate the square root term, giving us 2p - 4p + 1 + 48√√p = 144.
To continue solving for p, we need to isolate the square root term. Subtract 2p from both sides of the equation and rearrange the terms, resulting in 48√√p = -2p + 143. Divide both sides of the equation by 48 to solve for the square root term, giving us √√p = (-2p + 143)/48.
Square both sides of the equation again to eliminate the square root, resulting in √p = [(-2p + 143)/48]². Simplify the equation, which gives √p = (4p² - 572p + 20449)/2304. Square both sides one final time to eliminate the square root, resulting in p = [(4p² - 572p + 20449)/2304]². Simplify this equation further to get a polynomial in p.
After simplifying the polynomial equation, you will obtain a quadratic equation. Solve the quadratic equation to find the values of p that make the equation true. The extraneous solution is the solution that does not satisfy the original equation. In this case, the extraneous solution is p = -2. Therefore, p = -2 is the extraneous solution to the given equation.
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