Find the nature of roots of the equation 2x2-3x-1=0
Answers
Answer:
given 2x^2 - 3x - 1 = 0
to find nature of its roots
a = 2 , b= -3 and c= -1
if D : b^2 - 4ac = 0, real and equal roots
D: b^2 - 4ac > 0 , real and distinct roots
D: b^2 - 4ac < 0, imaginary roots
now on putting the values of a, b and c
we get D = (-3)^2 - 4(2)(-1)
= 9 - (-8)
= 17 > 0
hence roots are real and distinct
Answer:
Real and distinct
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 ;
2x² - 3x - 1 = 0 .
On comparing the given quadratic equation with the general quadratic equation ax² + bx + c = 0 ,
We have ;
a = 2
b = -3
c = -1
Thus,
The discriminant of the given quadratic equation will be given as ;
=> D = b² - 4ac
=> D = (-3)² - 4×2×(-1)
=> D = 9 + 8
=> D = 17
=> D > 0
Clearly ,
The discriminant of the given quadratic equation is greater than zero , thus both the roots of the given quadratic equation would be real and distinct .