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How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square?

Solve x2 - 4x + 5 = 0 by completing the square.

Summary

  1. A perfect square trinomial is a trinomial that can be factored into a binomial squared
  2. The constant term 'c' that will give us a perfect square trinomial is 4
  3. We can rewrite (x-2)(x-2) as (x-2)2
  4. The symbol above the (x-2)2 and the -1 is the square root symbol
  5. '±' means 'plus or minus'
  6. The square root of -1 is equal to the imaginary unit 'i'
  7. Since we had plus or minus an imaginary number, we end up with two complex solutions

Notes

    1. In order to create a perfect square trinomial, we need a constant value that will let us factor our trinomial into a binomial squared
    2. We can't do that when our constant is 5, so we're going to move it over so that we can start fresh!
    1. To move the 5 over, we need to subtract it from BOTH sides of the equation
    2. The 5s cancel out on the left and we're left with x2-4x
    3. 0-5 gives us -5 on the right
    1. The standard form of a quadratic equation is ax2+bx+c, where 'a', 'b', and 'c' are numbers and 'a' is not 0
    2. 'c', the number we're looking for, is the constant value at the end of the equation
    3. We want to find a value for 'c' that will let us factor the left hand side into a binomial squared
    1. We need to find a number to add to that so we can factor the trinomial into two identical binomials
    2. Since the first term of our trinomial is x2, the first term in each of our binomials will be x
    3. Since we know the signs have to be the same, and we need to get a -4x when we multiply the binomials, both our signs must be negative
    4. If we FOIL these binomials together, our two middle terms would be -2x, which combine to give us -4x
    5. So that means the second term in each binomial must be a 2
    6. This gives us a constant term of 4
    1. Remember, 'c' is the constant term in a quadratic: ax2+bx+c
    2. Divide the 'b' value by 2 - this will give you the second term in each of the binomials
    3. In our case, we got -2
    4. Then when you FOIL the binomials together, the constant you get at the end is the 'c' value
    1. If we add 4 to one side of the equation, we need to make sure to add it to the other side as well!
    2. That way we're sure to preserve the equality
    1. This should be easy, because we already have this trinomial factored from before!
    1. We used the factored form of the trinomial to figure out 'c'
    2. We can rewrite (x-2)(x-2) as (x-2)2
    1. Taking the square root of something squared will cancel out the square
    2. So we can get rid of the square on the left hand side by taking the square root
    3. But again, we need to make sure to do this on BOTH sides of the equation
    1. Squaring and taking the square root are opposite operations
    2. So taking the square root of something squared cancels out the square
    3. This leaves us with just x-2 on the left
    4. If you square a positive or a negative number, you always end up with a positive
    5. So we need to put a plus or minus (±) in front of the square root of -1 to account for both possibilities
    1. Taking the square root of a negative number gives us an imaginary number!
    2. The square root of -1 is defined as the imaginary unit 'i'
    1. Since we have a plus or minus, we have the possibility of two solutions
    2. So we need to split our equation into two equations so we can solve for both possibilities
    3. When we add 2 to both sides of each equation, we get the solutions x=2+i or x=2-i
    1. We can check our answers by plugging each of them into the original equation, x2-4x+5=0
    2. If we get true statements when we do this, we'll know our answers are correct!
    1. Remember, 'i' is the imaginary unit and is equal to the square root of -1
    2. Replace 'x' with '2+i' in the original equation, x2-4x+5=0
    3. (2+i)2 is the same as 2+i times 2+i
    4. We can FOIL complex numbers together the same way we FOIL two regular binomials
    5. We can also combine terms with 'i' just like we would like terms with the same variables
    6. i2 is equal to -1
    7. Since we get 0 when we combine all the like terms, we know our answer is correct!
    1. Remember, 'i' is the imaginary unit and is equal to the square root of -1
    2. Replace 'x' with '2-i' in the original equation, x2-4x+5=0
    3. (2-i)2 is the same as 2-i times 2-i
    4. We can FOIL complex numbers together the same way we FOIL two regular binomials
    5. We can also combine terms with 'i' just like we would like terms with the same variables
    6. i2 is equal to -1
    7. Since we get 0 when we combine all the like terms, we know our answer is correct!