determine the nature of the roots of the quadratic equation 5x^2 - 2 root 3 - 7
Answers
Question :
Determine the nature of the roots of the quadratic equation 5x² - 2√3 - 7 = 0 .
Answer :
Real and distinct
Solution :
- Given quadratic eq.: 5x² - 2√3 - 7 = 0
- To find : Nature of roots
The given quadratic equation can be rewritten as ;
=> 5x² - 2√3 - 7 = 0
=> 5x² - (2√3 + 7) = 0
=> 5x² + 0•x - (2√3 + 7) = 0
Now ,
Comparing the above quadratic equation with the general quadratic equation ax² + bx + c = 0 , we have ;
a = 5
b = 0
c = - (2√3 + 7)
Now ,
The discriminant will be given as ;
=> D = b² - 4ac
=> D = 0² - 4•5•[-(2√3 + 7)]
=> D = 20(2√3 + 7)
=> D > 0
Since ,
The discriminant of the given quadratic equation is greater than zero , thus its roots will be real and distinct .
Probable Question :
Determine the nature of the roots of the quadratic equation 5x² - 2√3x - 7 = 0 .
Answer :
Real and distinct
Solution :
- Given quadratic eq: 5x²- 2√3x -7 = 0
- To find : Nature of roots
Comparing the given quadratic equation with the general quadratic equation ax² + bx + c = 0 , we have ;
a = 5
b = -2√3
c = -7
Now ,
The discriminant will be given as ;
=> D = b² - 4ac
=> D = (-2√3)² - 4•5•(-7)
=> D = 12 + 140
=> D = 152
=> D > 0
Since ,
The discriminant of the given quadratic equation is greater than zero , thus its roots will be real and distinct .
Concept used :
★ 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
★ 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) .
We usually discuss the nature of the roots of the quadratic equation. However, the nature of the equation is that it is factorizable, with real roots such that
(5x+3)(x−1)=0
The graph crosses the x-axis twice at x=1 and x=−35 , which are its roots.
For this equation,
dydx=10x−2
So the equation has an extreme value at;
10x−2=0
So x=0.2
Small excursions around this value such as x=0.19 and x=0.21 show that the x=0.2 is a resting point (point of inflection) as the curve continues evolving.
d2ydx2=10 shows that the curve gets steeper at an increasing rate as x increases