Sign of a Quadratic Expression
A quadratic ax2 + bx + c keeps the sign of a for all x when D < 0, and changes sign only at its real roots.
What is Sign of a Quadratic Expression?
A quadratic ax2 + bx + c keeps the sign of a for all x when D < 0, and changes sign only at its real roots.
Key formula / rule: a > 0 with D < 0 gives an expression positive for every real x.
Key points
- Analyse the sign of a quadratic expression over the real line.
- Solve quadratic inequalities including parameterised cases.
Common exam trap
Multiplying an inequality by a variable expression of unknown sign.
Definitions
- Term
Sign of a Quadratic Expression
- Meaning
A quadratic ax2 + bx + c keeps the sign of a for all x when D < 0, and changes sign only at its real roots.
- Term
Sign of a Quadratic Expression — explanation
- Meaning
Reading a parabola's position relative to the x-axis converts an inequality into a statement about roots and the leading coefficient. This is the standard route for quadratic inequality problems.
Learning objectives
Analyse the sign of a quadratic expression over the real line.
Solve quadratic inequalities including parameterised cases.
Formulae
- Key point
a > 0 with D < 0 gives an expression positive for every real x.
- Key point
For D > 0 the expression takes the sign opposite to a strictly between the roots.
- Key point
The vertex value is c - b2/(4a), giving the extremum.
Prerequisites
MAT-U1-CQ-T2-S1-C2
Common mistakes
Multiplying an inequality by a variable expression of unknown sign.
Ignoring the endpoints when the inequality is non-strict.
Keywords
Sign
Quadratic
Expression
Practice preview
Which of the following statements about the quadratic expression ax^2 + bx + c is INCORRECT?…
medium
For what values of 'k' is the quadratic expression x^2 + 2x + k positive for all real values of x?…
easy
If the roots of the quadratic equation x^2 - 5x + 6 = 0 are alpha and beta, for what values of x is the expression x^2 - 5x + 6 > 0?…
easy