
[ended]Mathematics Tutoring Session #15 (Roots of a Quadratic Equation[Mon. 7th Mar 2016, 4:00  5:00pm])
Mathematics
Emmanuel Iwara(Tutor)
07032016 13:22:00 +0000Good day all!
This session has ended for today
The nature of the roots of a quadratic equation is often tested by jamb and other exams.So let quickly take a look at following.
Discriminant (D)  This is the expression under the square root in the quadratic formula. The discriminant determines the nature of the roots of a quadratic equation. The word '(nature)' refers to the types of numbers the roots can be — namely real, rational, irrational or imaginary.
x = b±√[(b^{2}  4ac) / 2a](1)
D = [(b^{2}  4ac) ](2)
(2) is called discriminant, which discriminates and analysis the nature of a root.
Nature of roots of a Quadratic Equation
When discriminant = b^{2} 4ac, then two cases arises
Case I.
When a, b, c are real numbers, a ≠0:
If = b^{2} 4 a c = 0, then roots are equal (and real).
If = b^{2} 4 a c > 0, then roots are real and unequal.
If = b^{2} 4 a c < 0, then roots are complex. It is easy to see that roots are a pair of complex conjugates.
Case II. When a, b, c are rational numbers, a ≠0:
If = b^{2} 4 a c = 0, then roots are rational and equal.
If = b^{2} 4 a c > 0, and is a perfect square of a rational number, then roots are rational and unequal.
If = b^{2} 4 a c > 0 but is not a square of rational number, then roots are irrational and unequal. They form a pair of irrational conjugates p +q, p  q where p, q Q, q> 0.
If =b^{2}  4 a c <0, then roots are a pair of complex conjugates.
Illustrative Examples
Example 1.
Discuss the nature of the roots of the following equations:
(i) 4x^{2}12x + 9 = 0
(ii) 3x^{2} 10x + 3 = 0
(iii) 9x^{2} 2 = 0
(iv) x^{2} + x + 1 = 0
Solution (i) b = 12, a = 4, c = 9
= b^{2} 4 a c = (12)^{2} 4 (4)(9) = 144 144 = 0.
Hence the roots are rational and equal.
(ii) b =10, a = 3, c = 3
= b^{2} 4 a c = (10)² 4 (3)(3) = 100 36 = 64.
Now = 64 > 0, and 64 is a perfect square of a rational number.
Hence the roots are rational and unequal.
(iii) b = 0, a = 9, c = 2
= b^{2} 4 a c = (0)^{2} 4 (9)(2) = 72. Now = 72 > 0 but is not a perfect square of a rational number. Hence the roots are irrational and unequal.
(iv) b= 1, a = 1, c = 1
= b^{2}  4 a c = (1)^{2} 4 (1)(1) = 3 < 0. Hence the roots are a pair of complex conjugates.
Example 2.
Find m so that the roots of the equation (4 + m)x^{2} + (m +1)x + 1 = 0 may be equal.
Solution
Using the formula b^{2}  4 a c = 0 (*)
a = (4 + m), b = (m + 1), c = 1. into equation (*)
b^{2} = 4ac
(m + 1)^{2} = 4(4+m)(1)
[m^{2} + 2m + 1] = 16 + 4m
m^{2} + 2m 4m + 1 16 = 0
m^{2}  2m 15 = 0(1)
using the formula method to solve for (m) in equation (1)
m = 3, 5.
Sum and Product of Roots
If the sum of a quadratic equation is given by β + α = b/a(1)
and the product as (β)(α) = c/a(2)
Then the quadratic equation can be written as x^{2} (β + α)x +(β)(α) = 0 (3)
Example 3. Find the equation whose roots are (3, 2)
Solution
Sum of roots = (3 + 2) = 1 = β + α
Product of the roots = (3)(2) = 6 = (α)(β)
Using equation (3), we have x^{2} (1)x + (6) = 0
= x^{2} + x  6 = 0
Now you may try to solve the following past jamb and other exams question;
Exercise 1. Find the nature of roots of the following equations.
(i) x^{2} + 9 = 0
(ii) x^{2} 22x + 1 = 0
2.For what value of k will the equation have equal root [x^{2} 2x + k 4] = 0
3.Find the equation whose roots are (5, 4)

{[reply.name]}
{[reply.voteCount]} {[reply.voteCount]} {[reply.created]}
{[reply.voteCount]} {[reply.created]}