find the value of k for which the quadratic equation x2 + 4x + k = 0 has real and distinct roots .
Answers
Question:
Find the value of k for which the quadratic equation x² + 4x + k = 0 has real and distinct roots.
Answer:
k € (-∞,4)
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 .
• A quadratic equation has atmost two roots .
• 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 ;
x² + 4x + k = 0
Clearly , we have ;
a = 1
b = 4
c = k
We know that ,
The quadratic equation will have real and distinct roots if its discriminant is greater than zero .
=> D > 0
=> 4² - 4•1•k > 0
=> 16 - 4•1•k > 0
=> 4(4 - k) > 0
=> 4 - k > 0
=> k < 4
=> k € (-∞,4)