In the following, determine whether the given quadratic equations have real roots and if so, find the roots:
(i)16x² = 24x + 1
(ii) x² + x + 2 = 0
(iii)√3x²+10x-8√3=0
Answers
SOLUTION :
(i) Given : 16x² = 24x + 1
16x² – 24x – 1 = 0
On comparing the given equation with, ax² + bx + c = 0
Here, a = 16 , b = - 24 , c = - 1
Discriminant , D = b² - 4ac
D = (-24)² - 4(16)(-1)
D = 576 + 64
D = 640
Since, D ≥ , 0 so given Quadratic equation has distinct real roots which are given by
x = [- b ± √D]/2a
x= −(−24) ±√640/ 2(16)
x =[ 24 ± 8√10] /32
x = 8 [3 ± √10]/32
x= (3 ± √10) /√4
x = (3 + √10) /√4 or x = (3 - √10) /√4
Hence, the Roots are (3 + √10) /√4 and x = (3 - √10) /√4.
(ii)Given : x² + x + 2 = 0
On comparing the given equation with, ax² + bx + c = 0
Here, a = 1 , b = 1 , c = 2
Discriminant , D = b² - 4ac
D = (1)² - 4(1)(2)
D = 1 - 8
D = - 7
Since, D < 0 so given Quadratic equation has no real roots .
(iii) Given : √3x² + 10x - 8√3 = 0
On comparing the given equation with, ax² + bx + c = 0
Here, a = √3 , b = 10 , c = - 8√3
Discriminant , D = b² - 4ac
D = (10)² - 4(√3)(- 8√3)
D = 100 + 32 × 3
D = 100 + 96
D = 196
Since, D ≥ , 0 so given Quadratic equation has distinct real roots which are given by
x = [- b ± √D]/2a
x= −10 ±√196/ 2(√3)
x =( - 10 ± 14)/2√3
x = 2( - 5 ± 7)/2√3
x = ( - 5 ± 7)/√3
x = (-5 + 7)/√3 or x = (- 5 - 7)/√3
x = 2/√3 or x = - 12/√3
x = - 12/√3 = (- 4 ×3)√3/√3×√3 = (- 4 ×3)√3 /3
[By rationalising the denominator]
x = - 4√3
x = 2/√3 or x = - 4√3
Hence, the Roots are 2/√3 and - 4√3 .
HOPE THIS ANSWER WILL HELP YOU...
Solution :
_______________________
Nature of roots of a
quadratic equation
ax²+bx+c=0, a≠0 is
Discreminant (D)
= b²-4ac
If i ) D > 0
Roots are real and distinct.
ii ) D = 0
Roots are real and equal.
iii ) D < 0
Roots are not real.
_____________________
(i)16x² = 24x + 1
=> 16x² - 24x - 1 = 0
Compare this with
ax²+bx+c =0 , we get
a = 16 , b = -24 , c = -1
D = b² - 4ac
= (-24)² - 4×16×(-1)
= 576 + 64
= 640
D > 0
Therefore ,
Roots are real and distinct.
(ii) x² + x + 2 = 0
a = 1 , b = 1 , c = 2
D = (1)² - 4×1×2
= 1 - 8
D = -7
D < 0
Therefore ,
Roots are not real.
(iii)√3x²+10x-8√3=0
a = √3 , b = 10, c = -8√3
D = (10)² - 4×(√3)×(-8√3)
= 100 + 96
= 196
D > 0
Therefore ,
Roots are real and distinct.
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