The Discriminant Explained: How Many Solutions Does a Quadratic Have?
PennPaper Team
The discriminant is the small expression hiding under the square root in the quadratic formula. Working it out first tells you how many solutions to expect, what kind they are, and whether the quadratic can be factored.
Die Idee in Kürze
For ax² + bx + c = 0, the discriminant is b² − 4ac. Positive means two real solutions, zero means one repeated solution, and negative means no real solutions.
Schritte
- Write the equation in standard form, ax² + bx + c = 0.
- Identify a, b, and c, keeping their signs.
- Calculate b² − 4ac carefully, using parentheses for negative values.
- Read the sign: positive gives two real roots, zero gives one, negative gives none.
Beispielaufgabe
How many real solutions does 3x² − 4x − 2 = 0 have?
Vollständige Lösung
Schritt für Schritt erklärt
Schritt 1
Here a = 3, b = −4, and c = −2. Substitute them into b² − 4ac.
Schritt 2
(−4)² is 16, and −4(3)(−2) is +24 because two negatives multiply to a positive.
Schritt 3
The discriminant is 40, which is positive, so there are two real solutions. Because 40 is not a perfect square, they are irrational.
Häufiger Fehler
The discriminant is b² − 4ac, not the square root of it. You only decide the number of solutions from its sign, so there is no need to take a square root at this stage.
Übungsaufgaben
- How many real solutions does x² + 4x + 4 = 0 have?
- How many real solutions does x² + x + 1 = 0 have?
- Find the discriminant of 2x² − 7x + 3 = 0 and describe the solutions.
Lösungen
- Discriminant 0: one repeated solution (x = −2)
- Discriminant −3: no real solutions
- Discriminant 25: two rational solutions (x = 3 and x = 1/2)
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