Learn the method, study an example, then explain your reasoning in an exam-style problem. Work on paper before opening the solution and compare each step with the marking guidance.
The discriminant and parameter conditions
Translate root and tangency conditions into inequalities, retaining all necessary restrictions.
Notes
For real coefficients and \(a\ne0\), the discriminant is \(\Delta=b^2-4ac\). A positive value gives two distinct real roots, zero gives a repeated real root, and a negative value gives no real roots.
The roots satisfy \(\alpha+\beta=-b/a\) and \(\alpha\beta=c/a\). Two distinct positive roots require \(\Delta>0\), a positive sum and a positive product. A positive discriminant alone does not establish the signs.
To study intersections, equate the two expressions for \(y\). A tangent corresponds to a repeated intersection root. If a parameter can make \(a=0\), handle that linear case separately.
Common mistakes
Using a non-strict inequality when roots must be distinct.
Treating two real roots as necessarily positive.
Ignoring a parameter value that turns a quadratic into a linear equation.
Three worked examples
Example 1
How many real roots does \(2x^2+4x+5=0\) have?
Show the reasoning
\(\Delta=4^2-4(2)(5)=-24<0\), so there are no real roots.
Example 2
Find \(k\) if \(x^2-6x+k=0\) has a repeated root.
Show the reasoning
\(36-4k=0\) gives \(k=9\). The equation is \((x-3)^2=0\), with repeated root \(3\).
Example 3
For which \(r\) does \(x^2-4x+r=0\) have two distinct positive roots?
Show the reasoning
The discriminant gives \(r<4\). The sum is \(4>0\); the product requires \(r>0\). Together, \(0