Find the value of k for which the quadratic polynomial 9x2-3kx+k has equal zeros
Answers
What's the condition for the quadratic equation ax² + bx + c = 0 if the equation has equal roots?
Well, the equation is nothing but a perfect square!
Let the equal roots of the equation be p. Then, sum of the roots,
2p = - b / a
p = - b / 2a → (1)
And the product of the roots,
p² = c / a
But from (1),
(- b / 2a)² = c / a
b² / 4a² = c / a
c = b² / 4a
Well, this is the condition for such a quadratic equation. The equation can also be derived by equating the discriminant to zero (This is possible since roots are equal).
But what's the need for it?!
Because, if our equation 9x² - 3kx + k = 0 is compared with ax² + bx + c = 0, we can see that c is replaced by k, so k can easily be found by applying the equation, can't it be?
Well,
a = 9 ; b = -3k ; c = k
Thus,
c = b² / 4a
k = (-3k)² / (4 × 9)
k = 9k² / 36
k = k² / 4
k² / k = 4
k = 4
Hence the value of k is 4. But wait, there's one more value for k too!
From the equation above,
k = k² / 4
k² = 4k
k² - 4k = 0
k(k - 4) = 0
This implies,
k = 0 ; k = 4
Thus we got the two values for k.
If k = 0,
9x² = (3x)²
If k = 4,
9x² - 12x + 4 = (3x - 2)²
#answerwithquality
#BAL
Answer:
no idea. the answer to these question