Positive and negative square roots

Key Notes :

1. What is a Square Root?

• A square root of a number is a value that, when multiplied by itself, gives the original number. It is represented by the radical symbol √.
• For example, the square root of 9 is 3 because 3 × 3 = 9.

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2. Positive and Negative Square Roots

• Every positive number has two square roots: one is positive and the other is negative.
  • For example, √9 = 3 and -√9 = -3 because both 3 × 3 = 9 and (-3) × (-3) = 9.

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3. The Square Root Symbol

• When a square root is written without specifying the sign, it refers to the positive square root.
  • For example, √16 = 4 (positive square root).
  • The negative square root is denoted by -√16 = -4.

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4. Zero as a Square Root

• The square root of zero is always zero.
  • √0 = 0

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5. Real Square Roots

• Square roots can only be taken for non-negative numbers in the real number system. This means square roots of negative numbers are not real numbers and are instead imaginary (denoted by ‘i’).
  • For example, √(-9) is an imaginary number because no real number multiplied by itself gives a negative result.

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6. Examples:

• √25 = 5 and -√25 = -5 (Both square roots of 25 are 5 and -5)
• √36 = 6 and -√36 = -6
• √49 = 7 and -√49 = -7

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7. Understanding Square Roots in Equations

• When solving equations involving square roots, consider both the positive and negative square roots.

• Example: If x² = 16, then x = √16 or x = -√16, so x = 4 or x = -4.

Learn with an example

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