(3k + 1)x^2+ 2(k + 1)x + k = 0 the equation has equal roots find k
Answers
b = 2 ( k + 1 )
c = k
b^2 - 4 ac = 0
4 k^2 +4 + 8k -4 ( 3k+1) (k )= 0
4 k^2 + 4 + 8 k -12k^2 -4k = 0
8k^2-4k - 4 = 0
8k^2 -8k + 4 k - 4 =0
8k ( k -1 ) + 4 ( k-1)= 0
So, k = 1 or -1/2
Answer:
k = 1 or k = (-1/2)
Step-by-step explanation:
We have,
(3k + 1)x² + 2(k + 1)x + k = 0
Solving,
(3k + 1)x² + 2(k + 1)x + k = 0
(3k + 1)x² + (2k + 2)x + k = 0
ax² + bx + c = 0
where a = (3k + 1), b = (2k + 2), c = k
Now, if a Quadratic equation must have two equal real roots then their Discriminants must be equal to zero.
So,
D = b² - 4ac = 0
Here, putting in the values,
(2k + 2)² - 4(3k + 1)(k) = 0
[(2k)² + (2 × 2k × 2) + (2)²] - 4(3k² + k) = 0
(4k² + 8k + 4) - 12k² - 4k = 0
4k² + 8k + 4 - 12k² - 4k = 0
-8k² + 4k + 4 = 0
Dividing the whole equation by (-2) we get,
4k² - 2k - 2= 0
a'k² + b'k + c' = 0
where a' = 4, b' = -2, c' = -2
We know that,
By Quadratic formula,
k = (-b' ± √(b'² - 4a'c'))/2a'
k = (-(-2) ± √((-2)² - 4(4)(-2)))/2(4)
k = (2 ± √(4 + 32))/8
k = (2 ± √36)/8
k = (2 ± 6)/8
k = (2 + 6)/8 or k = (2 - 6)/8
k = 8/8 or k = (-4)/8
k = 1 or k = (-1/2)
Hence,
k = 1 or k = (-1/2)
Hope it helped and believing you understood it........All the best