Question

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

(i) \(2x^2-3x+5=0\)

(ii) \(2x^2-6x+3=0\)

(iii) \(\frac{3}{5}x^2-\frac{2}{3}x+1=0\)

(iv) \(3x^2-4\sqrt{3}x+4=0\)

(v) \(3x^2-2\sqrt{6}x+2=0\)

Answer 1

(i) \(2x^2-3x+5=0\)

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

\(2x^2-3x+5=0\)

⇒ D = 9 – 4 × 5 × 2 = -31

Roots are not real.

(ii) \(2x^2-6x+3=0\)

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

\(2x^2-6x+3=0\)

⇒ D = 36 – 4 × 2 × 3 = 12 

Roots are real and distinct.

(iii) \(\frac{3}{5}x^2-\frac{2}{3}x+1=0\)

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

\(\frac{3}{5}x^2-\frac{2}{3}x+1=0\)

⇒ D = 4/9 – 4 × 3/5 × 1 = -88/45

Roots are not real.

(iv) \(3x^2-4\sqrt{3}x+4=0\)

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

\(3x^2-4\sqrt{3}x+4=0\)

⇒ D = 48 – 4 × 3 × 4 = 0

Roots are real and equal

(v) \(3x^2-2\sqrt{6}x+2=0\)

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

\(3x^2-2\sqrt{6}x+2=0\)

⇒ D = 24 – 4 × 3 × 2 = 0

Roots are real and equal.