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- The discriminant
The discriminant
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You can tell how many real roots a quadratic has without solving it. Just look at b2 − 4ac.
The discriminant
For ax2 + bx + c = 0, the discriminant is b2 − 4ac. It is the part under the square root in the quadratic formula.
What it tells you
b2 − 4ac > 0: two unequal real roots
b2 − 4ac = 0: two equal real roots (one repeated root)
b2 − 4ac < 0: no real roots
b2 − 4ac = 0: two equal real roots (one repeated root)
b2 − 4ac < 0: no real roots
Finding an unknown
kx2 + 6x + 1 = 0 has equal roots.
b2 − 4ac = 36 − 4k = 0, so k = 9.
For 'real roots' use ≥ 0. For 'no real roots' use < 0.
b2 − 4ac = 36 − 4k = 0, so k = 9.
For 'real roots' use ≥ 0. For 'no real roots' use < 0.
Find the values of k for which x2 + 2x + k = 0 has two unequal real roots.
- b2 − 4ac = 4 − 4k
- 4 − 4k > 0
- 4 > 4k
Answer: k < 1
b2 − 4ac > 0: two unequal real roots. = 0: equal roots. < 0: no real roots.
The interactive lesson includes the diagrams for this topic.
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