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How Do You Use The Discriminant to Determine the Number of Real or Complex Solutions to a Quadratic Equation?

Use the discriminant to determine if a quadratic equation has two real solutions, one real solution, or two complex solutions.

Summary

  1. Since there is no coefficient in front of x2, that means there is an invisible 1
  2. So 'a' for our equation is 1
  3. 'b', the coefficient in front of the 'x', is -2
  4. 'c', the constant term, is 3
  5. b2-4ac is the expression for the discriminant
  6. -8 is the value for our discriminant
  7. Since the discriminant is negative, our equation has two complex solutions

Notes

    1. The quadratic equation we're working with is x2-2x+3=0
    2. Remember, the discriminant tells us how many and what type of solutions a quadratic equation has
    1. The discriminant comes from the Quadratic Formula
    2. It tells us how many and what type of solutions a quadratic equation has
    1. Remember, the discriminant tells us how many and what type of solutions a quadratic equation has
    2. The symbol going over the b2-4ac is called a 'radical', or square root symbol
    3. The '±' symbol means 'plus or minus'
    4. The 'a', 'b', and 'c' values in the Quadratic Formula come from the general form of a quadratic: ax2+bx+c=0
    5. The discriminant is the expression under the radical in the Quadratic Formula: b2-4ac
    1. Remember, the discriminant is equal to b2-4ac
    2. The 'a', 'b', and 'c' values come from the general form of a quadratic equation: ax2+bx+c=0
    3. In the Quadratic Formula we have a plus and a minus in front of the radical
    4. This will give us two different answers, since we need to both add and subtract
    5. If we get 0 under the radical, we'd be adding and subtracting 0 since the square root of 0 is 0
    6. That would give us the same answer either way
    1. The standard form of a quadratic equation is ax2+bx+c=0
    2. Our quadratic equation is x2-2x+3=0
    3. Since there is no coefficient in front of x2, that means there is an invisible 1
    4. So 'a' for our equation is 1
    5. 'b', the coefficient in front of the 'x', is -2
    6. Remember to include the sign with the rest of the number!
    7. 'c', the constant term, is 3
    1. We need to plug our 'a', 'b', and 'c' values into the expression for the discriminant: b2-4ac
    2. For our equation a=1, b=-2, and c=3
    3. When we plug in for the discriminant we get (-2)2-4(1)(3)
    4. Squaring -2 gives us 4 and multiplying -4•1•3 gives us -12
    5. Taking 4-12 gives us -8, which is the value of our discriminant
    1. The discriminant is -8, so the number under the square root in the Quadratic Formula is negative
    2. The square root of a negative number gives us an imaginary number
    3. Since there is a plus or minus in front of the square root, we're both adding and subtracting an imaginary number
    4. So we will have two complex solutions to our equation