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Line 4:
Line 4: + + − All numbers have two square roots, although they don't have to be distinct. For example, the square root of zero has two values: +0 and -0.+ − + − −
→Two square roots for each number: rearranged slightly. removed +/- from the 9!
All numbers have two square roots that differ by a factor of -1:
All numbers have two square roots that differ by a factor of -1:
:<math>3 \cdot 3 = 9</math> and <math>-3 \cdot -3 = 9</math>
:<math>3 \cdot 3 = 9</math> and <math>-3 \cdot -3 = 9</math>
These two square roots don't have to be distinct. For example, the square root of zero has two values: +0 and -0. (While these may seem to be identical, in the mathematics of limits, they can have differing characteristics.)
When speaking of "''the'' square root of ''x''", people usually refer to the positive square root. For example:
When speaking of "''the'' square root of ''x''", people usually refer to the positive square root. For example:
:<math>\sqrt{9} = 3</math>
:<math>\sqrt{9} = 3</math>
Since in many situations only the positive square root has any meaning, to be clear that both square roots are to be considered, in mathematics the convention is to use a sign, pronounced "plus or minus" to signify the distinct possibilites:
:<math>\sqrt{9} = \pm 3</math>
Since in many situations only the positive square root has any meaning, to be clear that both square roots are to be considered, in mathematics the convention is to use a sign, pronounced "plus or minus", in front of the square root symbol:
:<math>\pm \sqrt{9} = \pm 3</math>
This has important implications when solving equations, in that, if the square root of both sides is taken, one must be careful to allow for both the positive and negative square root. A well known example is [[completing the square]] to solve a [[quadratic equation]].
This has important implications when solving equations, in that, if the square root of both sides is taken, one must be careful to allow for both the positive and negative square root. A well known example is [[completing the square]] to solve a [[quadratic equation]].