Consider the quadratic expression .

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Consider the quadratic expression

ax^{2}+bx+c\,(a\neq 0)

. (i) State the discriminant

\Delta

ax^{2}+bx+c

and write clearly the condition on

\Delta

for which the quadratic has (a) two distinct real roots, (b) equal real roots and (c) no real root. (ii) A student wishes to evaluate

\displaystyle \int \frac{dx}{ax^{2}+bx+c}

. Explain, with reasons based on the value of

\Delta

, whether the antiderivative obtained will involve a natural logarithm function or an inverse tangent function. (iii) Give one explicit example of each of the following and, without carrying out the complete integration, justify the nature of the final answer in the light of your explanation in part (ii): (A) an integral whose result will contain

\tan^{-1}

, (B) an integral whose result will contain

\ln|\;|

Related theory

Nature of Roots
Quadratic Equations
Integration Techniques
Integration by partial fractions
Integrals

Browse the problem library