Q. 15.0( 2 Votes )

Determine the nature of the roots of the following quadratic equations:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii) ,

(viii)

(ix)

Answer :

(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



D = 9 – 4 × 5 × 2 = -31


Roots are not 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 = 36 – 4 × 2 × 3 = 12


Roots are real and distinct.


(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 = 4/9 – 4 × 3/5 × 1 = -88/45


Roots are not real.


(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 = 48 – 4 × 3 × 4 = 0


Roots are real and equal


(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 = 24 – 4 × 3 × 2 = 0


Roots are real and 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



x2 – (2a + 2b)x + 4ab = 4ab


x2 – (2a + 2b)x = 0


D = (2a + 2b)2 – 0 = (2a + 2b)2


Roots are real and distinct


(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 = 576a2b2c2d2 – 4 × 16 × 9 × a2b2c2d2 = 0


Roots are real and equal


(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(a + b)2 – 4 × 2 × (a2 + b2)


D = -4(a2 + b2) + 2ab = -(a – b)2 – 3(a2 + b2)


Roots are not real


(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 = (a + b + c)2 – 4a(b + c)


D = a2 + b2 + c2 – 2ab – 2ac + 2bc


D = (a – b – c)2


Thus, roots are real and unequal


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