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Question
If the equation 9x2 + 6kx + 4 = 0 has equal roots, then the roots are both equal to
Options
± `2/3`
± `3/2`
0
±3
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Solution
Given, quadratic equation is: 9x2 + 6kx + 4 = 0
Now, the given equation has equal roots, so D = 0
⇒ b2 - 4ac = 0
⇒ (6k)2 - 4 × 9 × 4 = 0
⇒ 36k2 - 36 × 4 = 0
⇒ 36k2 = 36 × 4
⇒ k2 = 4
⇒ k = ± 2
Case 1: When k = 2, we have the quadratic equation as:
9x2 + 6(2)x + 4 = 0
⇒ 9x2 + 12x + 4 = 0
⇒ (3x)2 + 2 × 2 × 3x + (2)2 = 0
⇒ (3x + 2)2 = 0
⇒ 3x + 2 = 0
⇒ x = -`2/3`
Case 2: When k = -2, we have the quadratic equation as:
9x2 + 6(-2)x + 4 = 0
⇒ 9x2 - 12x + 4 = 0
⇒ (3x)2 - 2 × 2 × 3x + (2)2 = 0
⇒ (3x - 2)2 = 0
⇒ 3x - 2 = 0
⇒ x = `2/3`
So, two roots of the quadratic equation are `2/3 and -2/3 or ±2/3`.
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