# Determine the nature of the roots of the following quadratic equations:(i) (ii) (iii) (iv) (v) (vi) (vii)  , (viii) (ix) (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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