Quadratic Equations
The discriminant
Use the discriminant to tell how many real roots an equation has.
When splitting is hard, there is a formula that always works, the quadratic formula:
x = (–b ± √(b² – 4ac)) / 2a
The part under the root, D = b² – 4ac, is called the discriminant. It 'discriminates', meaning it tells the cases apart. A number can only be positive, zero or negative, so there are three cases:
| D = b² – 4ac | Roots |
|---|---|
| positive (> 0) | two different real roots |
| zero | two equal real roots, each –b/2a |
| negative (< 0) | no real roots (you can't take the square root of a negative number) |
Example: 2x² – 4x + 3 = 0 has D = 16 – 24 = –8, so it has no real roots.
So a quadratic equation has at most two roots, not always two. A common mistake is to say 'every quadratic has exactly two roots'. This equation has none.
'Real roots' means D ≥ 0: both 'different' and 'equal' count.
Finding k for equal roots: set D = 0 and solve for k. Then put each k back in and throw out any value that makes the x² number (a) zero, because then it is no longer quadratic.
'Is this possible?' Find D first. Two friends' ages add up to 20. Four years ago, their ages multiplied to 48. If one is x now, (x – 4)(16 – x) = 48, which gives x² – 20x + 112 = 0. D = 400 – 448 = –48, which is negative. No real x works, so the situation is impossible.
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