Ch. 7. Quadratic Equations and Expressions - Revision Points 1

Some definitions

Real polynomial

Complex polynomial

Degree of a polynomial

Polynomial equation

Some Results on roots of an equation


1. An equation of degree n has n roots, real or imaginary.

2. Surd and imaginary roots always occur in pairs, i.e. if 5-3i is a root of an equation, then 5 +3i is also its root. Similarly, if 3+SQRT(5) is a root of a given equation, then 3-SQRT(5) is also its root.

3. An odd degree equation has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of highest degree term is positive.

4. Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two real roots, one positive and one negative

3. Position of roots of a polynomial equation
4 Descartes rule of signs
5. Relations between roots and coefficients
6. Formation of a polynomial equation form given roots
7. Transformation of equations




8. Roots of a quadratic equation with real coefficients

ax²+bx+c where a≠0, a,b,c Є R is a quadratic equation with real coefficients.

The quantity D = b²-4ac is the called the discriminant of the quadratic equation.

1. The roots are real and distinct if and only if D>0.
2. The roots are real and equal if and only D = 0
3. The roots are complex with non-zero imaginary part if and only if D<0 .="" br="">4. The roots are rational iff a,b,c are rational and D is a proper square.
5. The roots are of the form p+√q (p,q Є Q), iff a,b,c are rational and D is not a perfrect square.
6. If a =1, b,c ЄI and the roots are rational numbers, then these roots must be integers.
7. If a quadratic equation in x has more than two roots, then it is an identity in x that is a=b=c=o.




9. Quadratic expression and its graph

Graph of a quadratic expression is a parabola.

10. Sign of a quadratic expression for real values of the variable
11. Solution of inequations
12. Position of roots of a quadratic equation
13. Common roots
14. Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
15. Condition for resolution into linear factors of a quadratic function
16. Algebraic interpretation of Rolle’s theorem




Updated 9 Jan 2016, 7 June 2008