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What is the Square Root Property?

What is the Square Root Property?

Summary

  1. 'C' represents any real number
  2. Taking a square root can give us either a positive or negative answer, so we need to account for both
  3. If C > 0, that means it is positive
  4. If C < 0, that means it is negative
  5. x2 = 25 has two real solutions: 5 and -5
  6. Since 0 is neither positive nor negative, 0 is the only answer we get when x2 = 0
  7. x2 = -81 has two imaginary solutions: 9i and -9i
  8. 'i' is the imaginary unit and is equal to the square root of -1

Notes

    1. Whenever you take the square root of something you have to account for both the positive and negative results
    2. 'C' represents any real number
    1. If 'C' is greater than 0, that means it will be positive
    2. Whenever 'C' is POSITIVE in x2=C, there will be TWO real number solutions
    3. 'x' is the variable we're trying to find solutions for
    1. Whenever 'C' is ZERO in x2=C, there will be ONE real number solution
    2. 'x' is the variable we're trying to find solutions for
    1. If 'C' is less than 0, that means it will be negative
    2. Whenever 'C' is NEGATIVE in x2=C, there will be two IMAGINARY number solutions
    3. Remember, imaginary numbers contain the imaginary unit 'i', which is equal to the square root of -1
    4. So if we take the square root of a negative, we'll end up with 'i's in our answer
    1. Remember, if 'C' is positive we should get two real solutions
    2. C is equal to 25
    3. The square and the square root cancel out on the left, leaving just 'x'
    4. On the right, we need the plus or minus (±) symbol to account for both the positive and negative square roots
    5. The square root of 25 could be either positive OR negative 5
    6. Since 5 and -5 are both real numbers, we get two real solutions
    1. Test the answers for the first case, x2 = 25
    2. The answers were 5 and -5
    3. To test the answers we got, plug them into the equation and see if we get a true statement
    4. Squaring 5 is the same as taking 5•5, which is 25
    5. Squaring -5 is the same as taking (-5)•(-5), which is also 25
    6. Both 5 and -5 give us 25 when we square them, so they are both solutions to x2 = 25
    1. We should get only one real solution when C=0
    2. The square and the square root cancel out on the left, leaving just 'x'
    3. On the right, the square root of 0 is just 0
    4. Zero is neither positive nor negative, so we get only one answer when we take the square root of 0
    1. Test the answer for the second case, x2 = 0
    2. The answer was 0
    3. To test the answer we got, plug it into the original equation and see if we get a true statement
    4. 02=0•0, which is definitely 0!
    5. So 0 is the only solution to x2 = 0
    1. Remember, if 'C' is negative we should get two imaginary solutions
    2. The square and the square root cancel out on the left, leaving just 'x'
    3. On the right, we need the plus or minus (±) symbol to account for both the positive and negative square roots
    4. We can factor a -1 out of -81
    5. Then use the Product Property of Square Roots to split up the square root
    6. The square root of -1 is equal to the imaginary unit 'i'
    7. We end up with two imaginary solutions: 9i and -9i
    1. Test the answers for the third case, x2 = -81
    2. The imaginary numbers +9i and -9i are the answers we just got
    3. 'i' is the imaginary unit and is equal to the square root of -1 -- it is NOT a variable
    4. Multiply 9•9 to get 81 and i•i to get i2
    5. i2 is always -1
    6. 9i is definitely a solution to x2 = -81
    7. Multiply (-9)•(-9) to get 81 and i•i to get i2
    8. -9i is also a solution to x2 = -81, so we have two imaginary solutions