find the value of k for which roots of the quadratic equations k x square - 14 x + 8 is equals to zero are equal
Answers
Let roots be α and β
A/q
α = 6β
now, if α and β are roots then equation will be (x -α)(x -β) =0
(x -α)(x -β) =0
⇒ x² - (α+β)x + αβ =0
now putting α = 6β ,
⇒x² - (6β +β)x + 6β×β =0
⇒x² - 7βx +6β² =0
now comparing with kx² -14x +8 =0
7β =14/k
⇒β =2/k
⇒β² = 4/k²_______(1)
and 6β² =8/k
⇒β² =4/3k_______(2)
equating (1) and (2), we get,
4/k² = 4/3k
k=3
Question:
Find the value of k for which the quadratic equation kx² - 14x + 8 = 0 has equal roots.
Answer:
k = 49/8
Note:
• An equation of degree 2 is know as quadratic equation .
• Roots of an equation is defined as the possible values of the unknown (variable) for which the equation is satisfied.
• The maximum number of roots of an equation will be equal to its degree.
• A quadratic equation has atmost two roots.
• The general form of a quadratic equation is given as , ax² + bx + c = 0 .
• The discriminant of the quadratic equation is given as , D = b² - 4ac .
• If D = 0 , then the quadratic equation would have real and equal roots .
• If D > 0 , then the quadratic equation would have real and distinct roots .
• If D < 0 , then the quadratic equation would have imaginary roots .
Solution:
The given quadratic equation is ;
kx² - 14x + 8 = 0
Clearly , we have ;
a = k
b = -14
c = 8
We know that ,
The quadratic equation will have equal roots if its discriminant is equal to zero .
=> D = 0
=> (-14)² - 4•k•8 = 0
=> 4•49 - 4•8k = 0
=> 4(49 - 8k) = 0
=> 49 - 8k = 0
=> 8k = 49
=> k = 49/8