Video Transcript
Solve the square root of 𝑥 minus
seven equals negative three.
When we’re working with equations
like these involving square roots, we usually solve them algebraically. However, any time we’re working
with square roots or any other type of roots, we have to be aware that we could
create extraneous solutions. This means we could find a value
for 𝑥 that doesn’t make this statement true.
On the left, we
have a square root of 𝑥 minus seven. And on the right, the value
negative three is a negative integer. This is a real number that is less
than zero, but the square root of 𝑥 minus seven can never produce a negative
integer because an integer is a real number. And if we have a value inside the
radical that is negative, it will produce imaginary solutions. And so we can say that the square
root of 𝑥 minus seven will never be equal to negative three.
By inspecting this equation
closely, we can say that this has no solution. But what if you didn’t recognize
this and tried to solve algebraically? You would have the square root of
𝑥 minus seven equals negative three. We then square both sides of the
equation, which gives us 𝑥 minus seven equals nine. If we add seven to both sides, we
get a statement that says 𝑥 equals 16. We’ve already said there’s no
possible solution. And this is why if you were solving
this algebraically, you would need to check to make sure that 𝑥 equals 16 is a
valid solution.
If we try to plug in 𝑥 equals 16
into this equation, we get the square root of 16 minus seven equals negative three,
that the square root of nine equals negative three. But the square root of nine is
three. And three is not equal to negative
three, which means 𝑥 cannot be equal to 16. It is an extraneous solution. And this equation has no
solution.
