Signs and quadratic inequalities
A factorised quadratic changes sign at simple roots. Find the roots, determine the sign in each interval and then include or exclude the roots according to the inequality symbol
\[f(x)=a(x-r_1)(x-r_2)\]
- Find the zeros Set the quadratic equal to zero to mark the boundary values on a number line \(\displaystyle f(x)=(x-2)(x-6)=0\Rightarrow x=2,6\)
- Test an interval Choose one value in each interval and evaluate the sign of the quadratic \(\displaystyle f(0)=(0-2)(0-6)=12>0\)
- Track the sign For an upward-opening quadratic with two simple roots, the signs are positive, negative, positive from left to right \(\displaystyle f(x):\quad +\text{ when }x<2,\ -\text{ when }2<x<6,\ +\text{ when }x>6\)
- Apply the endpoints Strict inequalities exclude zeros, while non-strict inequalities include them \(\displaystyle f(x)>0\Rightarrow x=2,6\text{ are excluded}\)
Worked example
Solve the quadratic inequality
\[-\left(x+1\right)\left(x-5\right)\ge0\]
- \(\displaystyle x=-1\text{ or }x=5\)The zeros are the boundary values where either factor is zero
- \(\displaystyle f(-2)=-7,\quad f(0)=5,\quad f(6)=-7\)Testing one value in each interval gives negative, positive, negative signs
- \(\displaystyle -1\le x\le5\)The non-strict inequality selects the positive interval and includes both zeros
Common mistakes
Reversing between and outside
✕\(\displaystyle 2<x<6\)✓\(\displaystyle x<2\text{ or }x>6\)
An upward-opening quadratic is positive outside two simple roots and negative between themIncluding strict endpoints
✕\(\displaystyle x\le2\text{ or }x\ge6\)✓\(\displaystyle x<2\text{ or }x>6\)
A strict inequality does not include the roots where the quadratic equals zeroChanging sign at a repeated root
✕\(\displaystyle -(x-3)^2\ge0\Rightarrow x\in\mathbb{R}\)✓\(\displaystyle -(x-3)^2\ge0\Rightarrow x=3\)
A squared factor is never negative, so its negative is at most zero and reaches zero only at the repeated rootDifficulty levels
introSolve a factorised inequality with integer roots by reading the sign intervals
standardSolve a quadratic inequality in general or factorised form and handle its endpoints
challengeTrack signs for repeated roots, non-monic quadratics or rearranged inequalities
examPresent a complete interval solution with correct endpoint inclusion for a quadratic inequality
Questions about this topic
Why are the roots placed on a number line?+
The roots divide the number line into intervals. The quadratic has a constant sign within each interval, so one test value identifies that interval
When are the roots included in the answer?+
Include a root for ≥ or ≤ because the quadratic may equal zero. Exclude it for > or < because the quadratic must be strictly positive or negative
Does the sign always change at a root?+
The sign changes at a simple root. At a repeated root the factor is squared, so the sign stays the same on both sides
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs
In this chapter
Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model