The discriminant, and lines meeting curves Cambridge IGCSE Additional Mathematics revision
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In plain words
A quadratic equation can have two answers, one answer or none. One small calculation, b² − 4ac, tells you which without solving anything. The same idea tells you whether a line crosses a curve, just touches it, or misses.
5 things to know
- For ax² + bx + c = 0, the discriminant is b² − 4ac.
- If b² − 4ac > 0, there are two different real roots. If b² − 4ac = 0, there are two equal roots. If b² − 4ac < 0, there are no real roots.
- To see how a line meets a curve, put the line's equation into the curve's to get a quadratic, and look at its discriminant.
- Discriminant > 0: the line crosses the curve at two points. Discriminant = 0: the line is a tangent. Discriminant < 0: the line does not meet the curve.
- Questions often ask for the values of a constant k that give one of these cases. Write the discriminant in terms of k and solve.
Worked example
Find the value of k for which the line y = x + k is a tangent to the curve y = x² + 3x + 2.
- Put them equal: x² + 3x + 2 = x + k, so x² + 2x + (2 − k) = 0.
- For a tangent, the discriminant is 0: 2² − 4 × 1 × (2 − k) = 0.
- 4 − 8 + 4k = 0, so k = 1.
Worked example
Find the values of k for which x² + kx + 9 = 0 has no real roots.
- The discriminant must be negative: k² − 36 < 0.
- So k² < 36, which gives −6 < k < 6.
Tips and tricks
- Rearrange the equation to "… = 0" before reading off a, b and c. Take care with signs: c may be an expression such as (2 − k).
- "Tangent" means equal roots. "Does not intersect" means no real roots. Translate the words into a condition on the discriminant.
It lands in your notebook with its questions as flashcards.
The discriminant, and lines meeting curves: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is the discriminant of x² + 6x + 9 = 0?
36 − 4 × 1 × 9 = 0, so the roots are equal.
What does b² − 4ac < 0 tell you?
The square root of a negative number is not real.
How many real roots does x² − 4x + 1 = 0 have?
The discriminant is 16 − 4 = 12, which is positive.
A line is a tangent to a curve. What is true of the discriminant of the equation formed from them?
The line and curve meet at one point, so the roots are equal.
For which value of k does 3x² + 6x + k = 0 have equal roots?
36 − 12k = 0.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 8 marks. Write your answers on paper, then check them.
The discriminant, and lines meeting curves
Cambridge IGCSE Additional Mathematics 0606 · 8 marks · papermunch.org
Name ______________________________ Date ______________
Find the discriminant of 2x² − 5x + 4 = 0 and state what it shows about the roots.[2]
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−7. It is negative, so there are no real roots. (25 − 32.)
Find the values of k for which x² + kx + 16 = 0 has two equal roots.[3]
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k = 8 or k = −8. k² − 64 = 0.
Show that the line y = 2x − 1 is a tangent to the curve y = x².[3]
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x² = 2x − 1 gives x² − 2x + 1 = 0. The discriminant is 4 − 4 = 0, so the roots are equal and the line is a tangent.
Answers: The discriminant, and lines meeting curves
- 1. −7. It is negative, so there are no real roots. (25 − 32.)
- 2. k = 8 or k = −8. k² − 64 = 0.
- 3. x² = 2x − 1 gives x² − 2x + 1 = 0. The discriminant is 4 − 4 = 0, so the roots are equal and the line is a tangent.