The discriminant
Objective. Classify the number of real roots using b²−4ac.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
The discriminant of a quadratic ax²+bx+c is the value b²−4ac, found inside the square root of the quadratic formula, and it tells you how many real solutions the equation has without actually solving it. If b²−4ac>0 there are two distinct real roots, if it equals 0 there is exactly one repeated root, and if it is negative there are no real roots at all. This connects directly to the graph: a positive discriminant means the parabola crosses the x-axis twice, zero means it touches once, and negative means it never meets the x-axis. For example, for x²−4x+4, the discriminant is (−4)²−4(1)(4)=0, confirming one repeated root.
Worked example
Find the discriminant of 3x²−2x+5 and state the number of real roots the equation has.
- 1Identify a=3, b=−2, c=5
- 2Compute b²: (−2)² = 4
- 3Compute 4ac: 4×3×5 = 60
- 4Discriminant = 4−60 = −56, which is negative
Answer: Discriminant = −56, so no real roots
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Practice
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Start practisingCommon mistake. Pupils often compute b²−4ac as (b−4ac)² or forget to square b before subtracting, miscalculating the discriminant of 2x²+3x−1 as 3−4(2)(−1)=11 instead of 3²−4(2)(−1)=17; always square b first.
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