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.




(ii)


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


(iii)


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


(iv)


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


(v)


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



(vi)


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




(vii)


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


(viii)


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


(ix)


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


(x)


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


(xi)


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


(xii)


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


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