Parameter conditions from a discriminant
First identify any parameter values that make the leading coefficient zero and solve those linear or constant cases separately. For the remaining quadratic cases, express the discriminant in terms of the parameter, find its critical values and apply the required inequality
- Check the degree Separate values with a(t)=0 before using the discriminant. A linear equation has one root if its x coefficient is nonzero; a constant equation has every real input as a solution when it is 0=0, and none otherwise \(\displaystyle tx^2+2x+1=0,\quad t=0\Rightarrow2x+1=0\Rightarrow x=-\frac12\)
- Form the quadratic Write the equation in ax² + bx + c = 0 form, keeping the parameter in its coefficient \(\displaystyle q_t(x)=a(t)x^2+b(t)x+c(t)=0\)
- Match the root condition Use equality for one repeated root, positivity for two distinct roots and negativity for no real roots \(\displaystyle \Delta=0\Rightarrow\text{one repeated root},\quad\Delta>0\Rightarrow\text{two distinct roots}\)
- Find critical values Solve Δ(t) = 0 to locate the parameter values where the root count can change \(\displaystyle \Delta(t)=(t-1)(t-9)=0\Rightarrow t=1\text{ or }t=9\)
- Test the intervals Use the sign of the factored discriminant between and outside its critical values \(\displaystyle \Delta(t)>0\Rightarrow t<1\text{ or }t>9\)
Worked example
For q_t(x) = x² + (t − 4)x + 1, find the values of t for which q_t(x) = 0 has two distinct real roots
- \(\displaystyle a=1,\quad b=t-4,\quad c=1\)Read the three coefficients from the quadratic
- \(\displaystyle \Delta(t)=(t-4)^2-4(1)(1)=t^2-8t+12\)Substitute the coefficients into the discriminant and expand
- \(\displaystyle \Delta(t)=(t-2)(t-6)\)Factorise the discriminant to identify its critical parameter values
- \(\displaystyle \Delta(t)>0\Rightarrow t<2\text{ or }t>6\)Two distinct real roots require a positive discriminant, which occurs outside the two critical values
Common mistakes
Difficulty levels
Questions about this topic
Why is Δ = 0 used for one repeated root?+
The quadratic formula has one value when the square-root part is zero. This occurs when the discriminant is zero, so the two roots coincide
How are parameter intervals found?+
Solve Δ(t) = 0 for the critical values, mark them on a number line and test the sign of the discriminant in each interval
Why must endpoint values be checked separately?+
First check whether the leading coefficient is zero and handle any linear or constant equation separately. Otherwise, a zero discriminant gives one repeated root, which is excluded when two distinct roots are required
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 3: Quadratic functions