# Determine the nature of the roots of the following quadratic equations: (i) (x - 2a)(x - 2b) = 4ab

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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