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!