If the quadratic equation x2 -5x + k = 0 has equal roots, then the value of k is ________________
Answers
Let p , q be roots of eqn
given p=q
sum of roots
=> p+q = -b/a
=> p+p = -(-5)
=> 2p = 5
=> p = 5/2
=> p² = 25/4
Now , consider , product
=> pq = c/a
=> p² = k
=> 25/4 = k
Answer :
k = 25/4
Note:
★ The possible values of the variable which satisfy the equation are called its roots or solutions .
★ A quadratic equation can have atmost two roots .
★ The general form of a quadratic equation is given as ; ax² + bx + c = 0
★ If α and ß are the roots of the quadratic equation ax² + bx + c = 0 , then ;
• Sum of roots , (α + ß) = -b/a
• Product of roots , (αß) = c/a
★ If α and ß are the roots of a quadratic equation , then that quadratic equation is given as : k•[ x² - (α + ß)x + αß ] = 0 , k ≠ 0.
★ The discriminant , D of the quadratic equation ax² + bx + c = 0 is given by ;
D = b² - 4ac
★ If D = 0 , then the roots are real and equal .
★ If D > 0 , then the roots are real and distinct .
★ If D < 0 , then the roots are unreal (imaginary) .
Solution :
Here,
The given quadratic equation is ;
x² - 5x + k = 0
Now ,
Comparing the given quadratic equation with the general quadratic equation
ax² + bx + c = 0 , we have ;
a = 1
b = -5
c = k
Also ,
It is given that , the roots of the given quadratic equation are equal .
Thus ,
The discriminant of the given quadratic equation must be equal to zero .
=> D = 0
=> b² - 4ac = 0
=> (-5)² - 4•1•k = 0
=> 25 - 4k = 0
=> 4k = 25
=> k = 25/4