Nature of the Roots
Relationship
between
Coefficients of Quadratic Equations
Roots of Quadratic Equations
Also known as . . .
Discriminants 𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎D
Discriminants
D
IfD is positive
The roots are
REALUNEQUAL&
Discriminants
D
If
D is equal to zero
The roots are
REAL EQUAL&
Discriminants
D
If
D is negative
The roots are
Imaginary UNEQUAL&
Discriminants
D
Moreover
D is a perfect square
The roots are
Rational
Discriminants
D
Moreover
D is not a perfect square
The roots are
Irrational
D is Zero, then roots are real and equal
𝑥2−6 𝑥+9=0 ,𝑎=1 ,𝑏=−6 ,𝑐=9
𝐷=𝑏2−4𝑎𝑐=(−6 )2−4 (1 ) (9 )¿36−36
¿0The roots are real, equal and rational
D is Zero, then roots are real and equal
𝑥2−6 𝑥+9=0 ,𝐷=0
𝑥=−(−6)±√−62−4 (1)(9)
2 (1)
𝑥1=3 𝑥2=3The roots are real, equal and rational
D is Positive, then roots are real and unequal
𝑥2+6 𝑥+5=0 ,𝑎=1 ,𝑏=+6 ,𝑐=5
𝐷=𝑏2−4𝑎𝑐=(6 )2−4 (1 ) (5 )¿36−20
¿16The roots are real, unequal and rational
D is Positive, then roots are real and Unequal
𝑥2+6 𝑥+5=0 ,𝐷=16
𝑥=−(6)±√62−4 (1)(5)
2(1)
𝑥1=−1 𝑥2=−5The roots are real, Unequal and rational
D is Negative, then roots are imaginary and unequal
2 𝑥2−4 𝑥+5=0 ,𝑎=2 ,𝑏=−4 ,𝑐=5
𝐷=𝑏2−4𝑎𝑐=(−4 )2−4 (2 ) (5 )¿16−40
¿−24The roots are imaginary and unequal
D is Positive, then roots are real and Unequal
2 𝑥2−4 𝑥+5=0 ,𝐷=−24
𝑥=−(−4)±√−42−4 (2)(5)
2(2)
𝑥1=1+ √64
The roots are imaginary and Unequal
𝑥2=1− √64
Thank You!