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Determine parameter values from a discriminant condition

Start a set of 8
Step 2 of 6

The family is x squared plus t times x plus 9. Move the slider for t across -8 to 8 and pause at -6 and 6. Watch the two bars and the discriminant readout to see when the bars match and when one bar is taller

Two bars for the family x squared plus t x plus 9: the b squared bar with value t squared and the 4ac bar fixed at 36, with readouts for both parts and the discriminant.Value050b^24ac036Part ofdiscriminant
b squared
0
Four a c
36
Discriminant
-36
0

For every position of the slider, where did the family have two, one or no real roots?

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, No calculatorBuild a worksheet

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

\[\Delta(t)=b(t)^2-4a(t)c(t)\]
  1. 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 \(\displaystyle tx^2+2x+1=0,\quad t=0\Rightarrow2x+1=0\Rightarrow x=-\frac12\)
  2. Form the quadratic Write the equation in ax² + bx + c = 0 form, keeping the parameter in its coefficient \(\displaystyle q_t(x)=a(t)x^2+b(t)x+c(t)=0\)
  3. Match the root condition Use equality for one repeated root, positivity for two distinct roots and negativity for no real roots \(\displaystyle \Delta=0\Rightarrow\text{one repeated root},\quad\Delta>0\Rightarrow\text{two distinct roots}\)
  4. Find critical values Solve Δ(t) = 0 to locate the parameter values where the root count can change \(\displaystyle \Delta(t)=(t-1)(t-9)=0\Rightarrow t=1\text{ or }t=9\)
  5. 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

\[q_t(x)=x^2+(t-4)x+1\]
  1. \(\displaystyle a=1,\quad b=t-4,\quad c=1\)Read the three coefficients from the quadratic
  2. \(\displaystyle \Delta(t)=(t-4)^2-4(1)(1)=t^2-8t+12\)Substitute the coefficients into the discriminant and expand
  3. \(\displaystyle \Delta(t)=(t-2)(t-6)\)Factorise the discriminant to identify its critical parameter values
  4. \(\displaystyle \Delta(t)>0\Rightarrow t<2\text{ or }t>6\)Two distinct real roots require a positive discriminant, which occurs outside the two critical values

Common mistakes

Using equality for two roots
✕\(\displaystyle \Delta=0\)✓\(\displaystyle \Delta>0\)
A zero discriminant gives one repeated root, whereas two distinct real roots require a positive discriminant
Reversing the sign intervals
✕\(\displaystyle 1<t<9\)✓\(\displaystyle t<1\text{ or }t>9\)
The factored discriminant is positive outside its two roots and negative between them
Including tangent endpoints
✕\(\displaystyle t\le1\text{ or }t\ge9\)✓\(\displaystyle t<1\text{ or }t>9\)
At t = 1 or t = 9 the discriminant is zero, so the roots are repeated rather than distinct

Difficulty levels

introUse a discriminant equality to find when a parameterised quadratic has one repeated root
standardSolve discriminant inequalities for at least one real root or no real roots
challengeHandle degree exclusions and definite-sign conditions involving the parameter
examCombine discriminant, interval and endpoint reasoning to satisfy a specified root condition

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

In this chapter

Factorise, solve and reconstruct a rational quadraticCalculate and interpret a quadratic discriminantTransform a quadratic and find its key featuresComplete the square and link equivalent formsDetermine a quadratic rule from exact informationSolve a quadratic exactly with the quadratic formulaConstruct a rational quadratic from a surd rootSolve an equation that reduces to quadratic formAnalyse signs and solve a quadratic inequalityIntersections of a line and a quadratic graphDetermine parameter values from a discriminant conditionFormulate and solve a quadratic problemAnalyse and optimise a quadratic model