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Find the values of k for which the roots are real and equal in each of the following equations:

(i) kx2 + 4x + 1 = 0

(ii) \(kx^2-2\sqrt{5}+4=0\) 

(iii) 3x2 - 5x + 2k = 0

(iv) 4x2 + kx + 9 = 0

(v) 2kx2 - 40x + 25 = 0

1 Answer

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Best answer

(i) kx2 + 4x + 1 = 0

For a quadratic equation, ax2 + bx + c = 0,

D = b2 – 4ac

If D = 0, roots are real and equal

kx2 + 4x + 1 = 0

⇒ D = 16 – 4k = 0

⇒ k = 4

(ii) \(kx^2-2\sqrt{5}x+4=0\)

For a quadratic equation, ax2 + bx + c = 0,

D = b2 – 4ac

If D = 0, roots are real and equal

\(kx^2-2\sqrt{5}x+4=0\)

⇒ D = 4 × 5 – 4 × 4k = 0

⇒ k = 5/4

(iii) 3x2 - 5x + 2k = 0

For a quadratic equation, ax2 + bx + c = 0,

D = b2 – 4ac 

If D = 0, roots are real and equal

3x2 - 5x + 2k = 0

⇒ D = 25 – 4 × 3 × 2k = 0

⇒ k = 25/24

(iv) 4x2 + kx + 9 =0

For a quadratic equation, ax2 + bx + c = 0,

D = b2 – 4ac

If D = 0, roots are real and equal

4x2 + kx + 9 =0

⇒ D = k2 – 4 × 4 × 9 = 0

⇒ k2 – 144 = 0

⇒ k = 12

(v) 2kx2 - 40x + 25 = 0

For a quadratic equation, ax2 + bx + c = 0

D = b2 – 4ac

If D = 0, roots are real and equal

2kx2 - 40x + 25 = 0

⇒ 1600 – 4 × 2k × 25 = 0 

⇒ k = 8

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