Question

16x²=24x+1
solve this question by quadratic formula {D=b²-4ac} & {x=[-b(+-)√D]/(2a)}.........
please solve this question.......​

Answer 1

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 .

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Answer 2

Answer:

$16 {x}^{2} - 24x - 1 = 0 \\ \\ \\ quadratic \: formula \: = \frac{ - b + \sqrt{ {b}^{2} - 4ac} }{2a} and \: \frac{ - b - \sqrt{ {b}^{2} - 4ac } }{2a} \\ \frac{24 + \sqrt{576 + 64} }{32} and \: \frac{24 - \sqrt{576 + 64} }{32} \\ \frac{24 + \sqrt{640} }{32} and \frac{24 - \sqrt{640} }{32} \\ \frac{24 + 8 \sqrt{10} }{32} and \frac{24 - 8 \sqrt{10} }{32} \\ \frac{8(3 + \sqrt{10}) }{32} and \frac{8(3 - \sqrt{10} )}{32} \\ \frac{3 + \sqrt{10} }{4} and \frac{3 - \sqrt{10} }{4}$

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