In the following determine whether the given quadratic equations have real roots and if so, find the roots:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(i)
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
16x2 – 24x – 1 = 0
⇒ D = 24 × 24 + 4 × 16 × 1 = 640
Roots are real.
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 1 – 4 × 2 = - 7
Roots are not real
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 100 + 4 × 8√3 × √3 = 196
Roots are real
⇒ x = (-10 � 14)/2√3
⇒ x = -4√3, 2/√3
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 4 – 4 × 2 × 3 = - 20
Roots are not real
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 4 × 6 – 4 × 3 × 2 = 0
Roots are equal
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 64a2b2 – 4 × 3a2 × 4b2 = 16a2b2
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
D = 20 + 4 × 5 × 3 = 80
⇒ x = -√5, √5/3
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 4 – 4 × 1 × 1 = 0
Roots are equal
x = (2 �√(4 – 4))/2 = 1
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 75 – 4 × 2 × 6 = 27
⇒ x = -2√3, -√3/2
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 49 – 4 × 5√2 × √2 = 9
⇒ x = -5/√2 , -√2
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
D = (2√2)2 – 4 × 2 × 1
⇒ D = 8 – 8 = 0
Roots are equal
x = (2√2 �0)/4
⇒ x = 1/√2, 1/√2
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 25 – 4 × 3 × 2 = 1
x = (5 � √1)/6
⇒ x = (5 � 1)/6
⇒ x = 1, 2/3