Quadratics and repeated roots
Use the discriminant to interpret how a quadratic meets the horizontal axis.
What you will learn
- Identify a, b and c correctly.
- Use the discriminant to find a parameter.
What the discriminant tells you
For ax²+bx+c=0 with a≠0, the quantity b²−4ac appears under the square root in the quadratic formula. A positive value produces two distinct real roots, zero produces one repeated real root, and a negative value produces no real roots.
A repeated root corresponds to the graph touching the x-axis at its vertex. This provides a geometric interpretation of setting the discriminant equal to zero.
Parameters are coefficients too
If one coefficient contains an unknown parameter, substitute it into b²−4ac just as you would a number. Solve the resulting equation or inequality and then check it against the requested root condition.
Worked example
Find k so that x²−10x+k=0 has equal roots.
Show the worked solution
- Identify a=1, b=−10, c=k.
- Set (−10)²−4k=0.
- Solve 100−4k=0; the resulting quadratic is (x−5)².
Answer k=25
Common mistakes
- Square the whole negative coefficient.
- Equal roots require Δ=0, not c=0.
What can you explain now?
How many distinct real roots does 2x²+3x+2=0 have?
Compare with the explanation
None
Its discriminant is 9−16=−7, which is negative.
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