determine the root of the following equations. Show the work to check and see if any of the roots are extraneous
√p/5-p=3
To determine the root of the equation √p/5 - p = 3, we can follow these steps:
Step 1: Isolate the radical term.
Add p to both sides of the equation:
√p/5 = 3 + p
Step 2: Square both sides of the equation to eliminate the radical.
(√p/5)^2 = (3 + p)^2
p/25 = 9 + 6p + p^2
Step 3: Multiply both sides of the equation by 25 to eliminate the fraction.
25(p/25) = 25(9 + 6p + p^2)
p = 225 + 150p + 25p^2
Step 4: Rearrange the equation into a quadratic form.
25p^2 + 150p + 225 - p = 0
25p^2 + 149p + 225 = 0
Step 5: Solve the quadratic equation.
Using the quadratic formula: p = (-b ± √(b^2 - 4ac)) / (2a), where a = 25, b = 149, and c = 225.
p = (-149 ± √(149^2 - 4*25*225)) / (2*25)
p = (-149 ± √(22201 - 22500)) / 50
p = (-149 ± √(-296)) / 50
The value inside the square root is negative, indicating that the equation has no real roots. Therefore, there are no roots to check for extraneous solutions.