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, ax^{2} + bx + c = 0,

D = b^{2} – 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, ax^{2} + bx + c = 0,

D = b^{2} – 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, ax^{2} + bx + c = 0,

D = b^{2} – 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, ax^{2} + bx + c = 0,

D = b^{2} – 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, ax^{2} + bx + c = 0,

D = b^{2} – 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, ax^{2} + bx + c = 0,

D = b^{2} – 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^{2} – (2a + 2b)x + 4ab = 4ab

⇒ x^{2} – (2a + 2b)x = 0

D = (2a + 2b)^{2} – 0 = (2a + 2b)^{2}

Roots are real and distinct

(vii) ,

For a quadratic equation, ax^{2} + bx + c = 0,

D = b^{2} – 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 = 576a^{2}b^{2}c^{2}d^{2} – 4 × 16 × 9 × a^{2}b^{2}c^{2}d^{2} = 0

Roots are real and equal

(viii)

For a quadratic equation, ax^{2} + bx + c = 0,

D = b^{2} – 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 × (a^{2} + b^{2})

⇒ D = -4(a^{2} + b^{2}) + 2ab = -(a – b)^{2} – 3(a^{2} + b^{2})

Roots are not real

(ix)

For a quadratic equation, ax^{2} + bx + c = 0,

D = b^{2} – 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 = a^{2} + b^{2} + c^{2} – 2ab – 2ac + 2bc

⇒ D = (a – b – c)^{2}

Thus, roots are real and unequal

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