If 2 is a root of the equation x²+ ax + 12 = 0 and the quadratic equation x² + ax + q = 0 has equal roots, then q =
(a)12
(b)8
(c)20
(d)16
Answers
SOLUTION :
Option (d) is correct : 16
Given : 2 is the root of x² + ax + 12 = 0 ………..(1)
and x² + ax + q = 0 has equal roots………(2)
Since, x = 2 is a root of equation (1) so it will satisfy the equation.
On putting x = 2 in equation (1)
x² + ax + 12 = 0
(2)² + 2a + 12 = 0
4 + 2a + 12 = 0
16 + 2a = 0
2a = - 16
a = -16/2
a = - 8
On putting a = - 8 in equation (2)
x² + ax + q = 0
x² + (- 8)x + q = 0
x² - 8x + q = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 1 , b = - 8 , c = q
D(discriminant) = b² – 4ac
D = (- 8)² - 4(1)(q)
D = 64 - 4q
D = 0 ( Equal roots given)
64 - 4q = 0
64 = 4q
q = 64/4
q = 16
Hence the value of q is 16 .
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Answer:
(d) 16
Step-by-step explanation:
In case it helps, you might like to see the second step done a bit differently.
As in the other solution, put x = 2 in the first equation:
4 + 2a + 12 = 0 => a = -8
In the second equation, the sum of the roots is -a / 1 = -a = 8. Since the roots are equal, they are both 4. Therefore their product is q = 4 × 4 = 16.