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
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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
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
Find the values of k for which the roots are real and equal in each of the following equations:
(i) 2x2 + kx + 3 = 0
(ii) kx(x – 2) + 6 = 0
(iii) x2 – 4kx + k = 0
(iv) \(k\text{x}(\text{x}-2\sqrt{5})+10=0\)
(v) px(x – 3) + 9 = 0
(vi) 4x2 + px + 3 = 0
Find the values of k for which the roots are real and equal in each of the following equations:
(i) 2x2 + kx + 3 = 0
(ii) kx(x – 2) + 6 = 0
(iii) x2 – 4kx + k = 0
(iv) \(k\text{x}(\text{x}-2\sqrt{5})+10=0\)
(v) px(x – 3) + 9 = 0
(vi) 4x2 + px + 3 = 0
(i) 2x2 + kx + 3 = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
2x2 + kx + 3 = 0
⇒ D = k2 – 4 × 2 × 3 = 0
⇒ k2 = = 24
⇒ k = 2√6
(ii) kx(x – 2) + 6 = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
kx(x – 2) + 6 = 0
⇒ kx2 – 2kx + 6 = 0 ⇒ D
= 4k2 – 4 × 6 × k = 0
⇒ 4k(k – 6) = 0
⇒ k = 0, 6 but k can’t be 0 a it is the coefficient of x2, thus k = 6
(iii) x2 – 4kx + k = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
x2 – 4kx + k = 0
⇒ D = 16k2 – 4k = 0
⇒ 4k(4k – 1) = 0
⇒ k = 0, 1/4
(iv) \(k\text{x}(\text{x} – 2\sqrt{5})+10=0\)
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
\(k\text{x}(\text{x} – 2\sqrt{5})+10=0\)
⇒ kx2 – 2√5kx + 10 = 0
⇒ D = 4 × 5k2 – 4 × k × 10 = 0
⇒ k2 = 2k
⇒ k = 2
(v) px(x – 3) + 9 = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
px(x – 3) + 9 = 0
⇒ px2 – 3px + 9 = 0
⇒ D = 9p2 – 4 × 9 × p = 0
⇒ p = 4
(vi) 4x2 + px + 3 = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
4x2 + px + 3 = 0
⇒ D = p2 – 4 × 4 × 3 = 0
⇒ p2 = 48
⇒ p = 4√3
Find the values of k for which the roots are real and equal in each of the following equations:
(i) 9x2 – 24x + k = 0
(ii) 4x2 – 3kx + 1 = 0
(iii) x2 – 2(5 + 2k)x+3(7 + 10k) = 0
(iv) (3k + 1) x2 + 2 (k + 1) x + k = 0
(v) kx2 + kx + 1 = -4x2 – x
Find the values of k for which the roots are real and equal in each of the following equations:
(i) 9x2 – 24x + k = 0
(ii) 4x2 – 3kx + 1 = 0
(iii) x2 – 2(5 + 2k)x+3(7 + 10k) = 0
(iv) (3k + 1) x2 + 2 (k + 1) x + k = 0
(v) kx2 + kx + 1 = -4x2 – x
(i) 9x2 – 24x + k = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
9x2 – 24x + k = 0
⇒ D = 576 – 4 × 9 × k = 0
⇒ k = 576/36 = 16
(ii) 4x2 – 3kx + 1 = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
4x2 – 3kx + 1 = 0
⇒ D = 9k2 – 4 × 4 × 1 = 0
⇒ 9k2 = 16
⇒ k = 4/3
(iii) x2 – 2(5 + 2k)x + 3(7 + 10k) = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
x2 – 2(5 + 2k)x + 3(7 + 10k) = 0
⇒ D = 4(5 + 2k)2 – 4 × 3(7 + 10k) = 0
⇒ 100 + 16k2 + 80k – 84 – 120k = 0
⇒ 16k2 – 40k + 16 = 0
⇒ 2k2 – 5k + 2 = 0
⇒ 2k2 – 4k – k + 2 = 0
⇒ 2k(k – 2) – (k – 2) = 0
⇒ (2k – 1)(k – 2) = 0
⇒ k = 2, 1/2
(iv) (3k + 1) x2 + 2 (k + 1) x + k = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
(3k + 1) x2 + 2 (k + 1) x + k = 0
⇒ D = 4(k + 1)2 – 4k(3k + 1) = 0
⇒ 4k2 + 8k + 4 – 12k2 – 4k = 0
⇒ 2k2 – k – 1 = 0
⇒ 2k2 – 2k + k – 1 = 0
⇒ 2k(k – 1) + (k – 1) = 0
⇒ (2k + 1)(k – 1) = 0
⇒ k = 1, -1/2
(v) kx2 + kx + 1 = -4x2 – x
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
kx2 + kx + 1 = -4x2 – x
⇒ (k + 4)x2 + (k + 1)x + 1 = 0
D = (k + 1)2 – 4(k + 4) = 0
⇒ k2 + 2k + 1 – 4k – 16 = 0
⇒ k2 – 2k – 15 = 0
⇒ k2 – 5k + 3k – 15 = 0
⇒ k(k – 5) + 3(k – 5) = 0
⇒ (k + 3)(k – 5) = 0
⇒ k = 5, -3
Find the values of k for which the roots are real and equal in each of the following equations:
(i) (k + 1) x2 + 2 (k + 3) x + (k + 8) = 0
(ii) x2 – 2kx + 7x + 1/4 = 0
(iii) (k + 1) x2 – 2 (3k + 1) x + 8k + 1 = 0
(iv) 5x2 – 4x + 2 +k ((4x2 – 2x – 1) = 0
(v) (4 – k) x2 + (2k + 4) x + (8k + 1) = 0
Find the values of k for which the roots are real and equal in each of the following equations:
(i) (k + 1) x2 + 2 (k + 3) x + (k + 8) = 0
(ii) x2 – 2kx + 7x + 1/4 = 0
(iii) (k + 1) x2 – 2 (3k + 1) x + 8k + 1 = 0
(iv) 5x2 – 4x + 2 +k ((4x2 – 2x – 1) = 0
(v) (4 – k) x2 + (2k + 4) x + (8k + 1) = 0
(i) (k + 1) x2 + 2 (k + 3) x + (k + 8) = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
(k + 1) x2 + 2 (k + 3) x + (k + 8) = 0
⇒ D = 4(k + 3)2 – 4(k + 1)(k + 8) = 0
⇒ 4k2 + 36 + 24k – 4k2 – 32 – 36k = 0
⇒ 12k = 4
⇒ k = 1/3
(ii) x2 – 2kx + 7x + 1/4 = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
x2 – 2kx + 7x + 1/4 = 0
⇒ D = (7 – 2k)2 – 4 × 1/4 = 0
⇒ 49 + 4k2 – 28k – 1 = 0
⇒ k2 – 7k + 12 = 0
⇒ k2 – 4k – 3k + 12 = 0
⇒ k(k – 4) – 3(k – 4) = 0
⇒ (k – 3)(k – 4) = 0
⇒ k = 3, 4
(iii) (k + 1)x2 – 2(3k + 1)x + 8k + 1 = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
(k + 1)x2 – 2(3k + 1)x + 8k + 1 = 0
⇒ D = 4(3k + 1)2 – 4(k + 1)(8k + 1) = 0
⇒ 4 × (9k2 + 6k + 1) – 32k2 – 4 – 36k = 0
⇒ 36k2 + 24k + 4 – 32k2 – 4 – 36k = 0
⇒ 4k(k – 3) = 0
⇒ k = 0, 3
(iv) 5x2 – 4x + 2 + k (4x2 – 2x – 1) = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
5x2 – 4x + 2 + k (4x2 – 2x – 1) = 0
⇒ (5 + 4k)x2 – (4 + 2k)x + 2 – k = 0
⇒ D = (4 + 2k)2 – 4 × (5 + 4k)(2 – k) = 0
⇒ 16 + 4k2 + 16k + 16k2 – 12k – 40 = 0
⇒ 20k2 – 4k – 24 = 0
⇒ 5k2 – k – 6 = 0
⇒ 5k2 – 6k + 5k – 6 = 0
⇒ k(5k – 6) + (5k – 6) = 0
⇒ (k + 1)(5k – 6) = 0
⇒ k = -1, 6/5
(v) (4 – k)x2 + (2k + 4)x + (8k + 1) = 0
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are real and equal
(4 – k)x2 + (2k + 4)x + (8k + 1) = 0
⇒ D = (2k + 4)2 – 4 × (4 – k)(8k + 1) = 0
⇒ 4k2 + 16 + 16k + 32k2 – 16 – 124k = 0
⇒ 36k2 – 108k = 0
⇒ 36k(k – 3) = 0
⇒ k = 0, 3
(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