How do you solve -4 sqrt(x+9) = 20 and find any extraneous solutions?

1 Answer
Stefan V.
Aug 3, 2016

x in O/

Explanation:

Your radical equation has no solutions for real numbers. Here's why.

Isolate the square root on one side of the equation by dividing both sides by -4

(color(red)(cancel(color(black)(-4))) * sqrt(x+9))/color(red)(cancel(color(black)(-4))) = 20/(-4)

sqrt(x+9) = -5

By definition, the square root of a positive real number must always produce a positive value. This implies that you must have

  • x + 9 >= 0 -> you must take the square root of a positive number when working in RR

  • color(red)(cancel(color(black)(-5 >=0))) -> the square root of a positive number must be a positive number

In your case ,-5 is clearly not positive, which implies that your equation has no solutions when working in RR. Simply put, you can't find a value of x in RR that can satisfy both conditions.

You can have x+9 >=0 for x in [-9, +oo), but you will never have -5 >=0, regardless of the value of x in [-9, + oo).

You can write this as

x in O/ -> no solutions in RR

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