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Calculate and interpret a quadratic discriminant

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Discriminant and root type

For a quadratic with real coefficients and a non-zero leading coefficient, the sign of the discriminant gives the number of distinct real roots. When the coefficients are rational and the discriminant is positive, the roots are rational exactly when it is the square of a rational number; otherwise both roots are irrational

\[ax^2+bx+c=0,\quad a\ne0;\qquad\Delta=b^2-4ac\]
  1. 1Use zero form Write the quadratic with zero on one side and a non-zero leading coefficient, then identify its three signed coefficients \(\displaystyle x^2-8x+20=0\Rightarrow a=1,\;b=-8,\;c=20\)
  2. 2Square b Place a negative b in brackets before squaring, then subtract four times a times c \(\displaystyle b=-8\Rightarrow b^2=(-8)^2=64\)
  3. 3Classify the sign A positive discriminant means two distinct real roots, a zero discriminant means one repeated real root, and a negative discriminant means no real roots \(\displaystyle \Delta=64-4(1)(20)=-16<0\Rightarrow\text{no real roots}\)
  4. 4Check square status With rational coefficients and a positive discriminant, test whether the discriminant is a rational square. A reduced positive fraction is a rational square exactly when its numerator and denominator are both integer squares \(\displaystyle \Delta=\frac14=\left(\frac12\right)^2\Rightarrow\text{rational roots},\quad\Delta=24\Rightarrow\text{irrational roots}\)
Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, No calculatorBuild a worksheet

Discriminant and root type

For a quadratic with real coefficients and a non-zero leading coefficient, the sign of the discriminant gives the number of distinct real roots. When the coefficients are rational and the discriminant is positive, the roots are rational exactly when it is the square of a rational number; otherwise both roots are irrational

\[ax^2+bx+c=0,\quad a\ne0;\qquad\Delta=b^2-4ac\]
  1. Use zero form Write the quadratic with zero on one side and a non-zero leading coefficient, then identify its three signed coefficients \(\displaystyle 2x^2-5x=7\Rightarrow2x^2-5x-7=0\)
  2. Square b The linear coefficient is squared, so put the entire signed value in brackets before squaring \(\displaystyle b=-5\Rightarrow b^2=(-5)^2=25,\quad4ac=4(2)(-7)=-56\)
  3. Classify the sign The sign of the discriminant tells whether the quadratic has two, one or no real roots \(\displaystyle \Delta>0\Rightarrow2\text{ real roots},\quad\Delta=0\Rightarrow1\text{ repeated root},\quad\Delta<0\Rightarrow\text{no real roots}\)
  4. Check square status With rational coefficients and a positive discriminant, test whether the discriminant is a rational square. A reduced positive fraction is a rational square exactly when its numerator and denominator are both integer squares \(\displaystyle \Delta=\frac14=\left(\frac12\right)^2\Rightarrow\text{rational roots},\quad\Delta=24\Rightarrow\text{irrational roots}\)

Worked example

For the displayed quadratic equation, calculate the discriminant and state the type of its roots

\[3x^2+2x-1=0\]
  1. \(\displaystyle a=3,\quad b=2,\quad c=-1\)Read the coefficients from the zero-form equation
  2. \(\displaystyle \Delta=b^2-4ac=2^2-4(3)(-1)=4+12=16\)Substitute the coefficients into the discriminant formula
  3. \(\displaystyle \Delta=16>0\)A positive discriminant means that there are two distinct real roots
  4. \(\displaystyle 16=4^2\Rightarrow\text{the roots are rational}\)The coefficients are rational and the positive discriminant is a rational square, so both real roots are rational

Common mistakes

Failing to square b
✕\(\displaystyle \Delta=-5-4(2)(-3)=19\)✓\(\displaystyle \Delta=(-5)^2-4(2)(-3)=49\)
The formula squares the linear coefficient, so square its complete signed value
Reversing the sign classification
✕\(\displaystyle \Delta=-4\Rightarrow\text{two distinct real roots}\)✓\(\displaystyle \Delta=-4\Rightarrow\text{no real roots}\)
A negative discriminant means that the quadratic has no real roots
Calling every positive discriminant rational
✕\(\displaystyle \Delta=12\Rightarrow\text{two rational roots}\)✓\(\displaystyle \Delta=12\Rightarrow\text{two irrational roots}\)
With rational coefficients, a positive discriminant that is not a rational square gives two irrational real roots

Difficulty levels

introUse integer coefficients in zero form to calculate Δ and classify roots, real factors or horizontal intercepts
standardHandle signed integer coefficients, including equations that require rearrangement, and classify using Δ
challengeUse rational coefficients or justify a constant sign; distinguish rational squares from other positive discriminants
examLink coefficients and Δ to root, factor or intercept classification; some cases also ask for the sign for all real x

Questions about this topic

Why must the equation be written with zero on one side?+

The discriminant uses the signed coefficients of the quadratic after every term is on one side and the other side is zero; rearranging first prevents coefficient-sign errors

What does a zero discriminant mean?+

It means there is one repeated real root. The quadratic touches the x-axis at one point

When are the roots irrational?+

For a quadratic with rational coefficients, if Δ is positive but is not the square of a rational number, both real roots are irrational

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