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

(i) \((x-2a)(x-2b)=4ab\) 

(ii) \(9a^2b^2x^2-24abcdx+16c^2d^2=0,a\neq0,b\neq0\) 

(iii) \(2(a^2+b^2)x^2+2(a+b)x+1=0\)

(iv) \((b+c)x^2-(a+b+c)x+a=0\)

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(i) \((x-2a)(x-2b)=4ab\)

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

\((x-2a)(x-2b)=4ab\)

⇒ x2 – (2a + 2b)x + 4ab = 4ab 

⇒ x2 – (2a + 2b)x = 0 

D = (2a + 2b)2 – 0 = (2a + 2b)2

Roots are real and distinct

(ii) \(9a^2b^2x^2-24abcdx+16c^2d^2=0,a\neq0,b\neq0\)

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

\(9a^2b^2x^2-24abcdx+16c^2d^2=0,a\neq0,b\neq0\)

⇒ D = 576a2b2c2d2 – 4 × 16 × 9 × a2b2c2d2 = 0

Roots are real and equal

(iii) \(2(a^2+b^2)x^2+2(a+b)x+1=0\)

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

\(2(a^2+b^2)x^2+2(a+b)x+1=0\)

⇒ D = 4(a + b)2 – 4 × 2 × (a2 + b2)

⇒ D = 4(a + b)2 – 4 × 2 × (a2 + b2)

Roots are not real

(iv) \((b+c)x^2-(a+b+c)x+a=0\)

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

\((b+c)x^2-(a+b+c)x+a=0\)

⇒ 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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