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Solving Quadratic Equations – The Discriminant
• The Discriminant is the expression found under the radical symbol in the quadratic formula.
• Discriminant =
2 4
2
b b acx
a
2 4b ac
Three possible solutions of a quadratic equation.
Two Different Real Solutions
2 4 0b ac • Case I: (discriminant is positive)
2 10
2x
Example:
2 10 2 10,
2 2x
Discriminant
One Repeated Real Solution
2 4 0b ac • Case II: (discriminant is zero)
2 0
2x
Example:
1, 1x 2 0 2 0,
2 2x
Two Different Imaginary Solutions
2 4 0b ac • Case III: (discriminant is negative)
2 4
2x
Example:
1 , 1x i i 2 2
2
ix
Two Different Imaginary Solutions
2 4 0b ac • Case III:
2 4 0b ac • Case I:
Two Different Real Solutions
2 4 0b ac • Case II:
One Repeated Real Solution
Summary
• Example 1:
Move all terms to left side
Determine the type and number of solutions for the quadratic equation.
2 12x x
2 12 0x x
Determine the values of a, b, and c
1, 1, 12a b c
Determine the value of the discriminant
1, 1, 12a b c
21 4(1)( 12) 2 4b ac
1 48 49 0
Two Different Real Solutions