Question

find whether the following equations have real roots. if real roots exist,find them. (i) 8x^2 + 2x - 3=0 (ii) -2x^2 + 3x + 2=0 (iii) 5x^2 - 2x - 10=0 (iv) 1 / 2x-3 + 1 / x-5=1 , x ≠ 3 / 2 , 5 (v) x^2 + 5√5 - 70=0

Answer 1

$i) Compare \: given \: Quadratic \: equation$

$8x^{2}+2x-3 = 0 \: with \: ax^{2}+bx+c=0 ,we$

$get$

$a = 8 , \: b = 2 \:and \: c = -3$

$Discreminant (D)$

$= b^{2} - 4ac$

$= 2^{2} - 4\times 8 \times (-3)$

$= 4 + 96$

$= 100 \gt 0$

$\therefore \green { Roots \: are \: real \:and \: distinct . }$

$ii) Compare \: given \: Quadratic \: equation$

$-2x^2 + 3x + 2=0 \: with \: ax^{2}+bx+c=0 ,we$

$get$

$a = -2 , \: b = 3 \:and \: c = 2$

$Discreminant (D)$

$= b^{2} - 4ac$

$= 3^{2} - 4\times (-2) \times 2$

$= 9 + 16$

$= 25 \gt 0$

$\therefore \green { Roots \: are \: real \:and \: distinct . }$

$iii) Compare \: given \: Quadratic \: equation$

$5x^2 -2 x - 10 =0 \: with \: ax^{2}+bx+c=0 ,we$

$get$

$a = 5 , \: b = -2 \:and \: c = -10$

$Discreminant (D)$

$= b^{2} - 4ac$

$= (-2)^{2} - 4\times 5 \times (-10)$

$= 4 + 200$

$= 204 \gt 0$

$\therefore \green { Roots \: are \: real \:and \: distinct . }$

$iv) Given \: Quadratic \: equation :$

$\frac{1}{2x-3} + \frac{1}{x-5} = 1$

$\implies \frac{x-5+2x-3}{(2x-3)(x-5)} = 1$

$\implies \frac{ 3x-8 }{2x^{2} - 10x - 3x + 15} = 1$

$\implies \frac{ 3x-8 }{2x^{2} - 13x + 15} = 1$

$\implies 3x - 8 = 2x^{2} - 13x + 15$

$\implies 0 = 2x^{2} - 13x + 15- 3x + 8$

$\implies 2x^{2} - 16x + 23 = 0$

$Compare \: given \: Quadratic \: equation$

$2x^2 -16x +23 =0 \: with \: ax^{2}+bx+c=0 ,we$

$get$

$a = 2 , \: b = -16\:and \: c = 23$

$Discreminant (D)$

$= b^{2} - 4ac$

$= (-16)^{2} - 4\times 2 \times 23$

$= 256 - 184$

$= 72 \gt 0$

$\therefore \green { Roots \: are \: real \:and \: distinct . }$

•••♪

Answer 2

$\sf\red{ i) \ Compare \: given \: Quadratic \: equation }$

$\sf\purple{ 8x^{2}+2x-3 = 0 \: with \: ax^{2}+bx+c=0 , \ we \ get}$

$\sf \purple{a = 8 , \: b = 2 \:and \: c = -3}$

$\sf\purple{ Discreminant (D)}$

$\sf \purple{= b^{2} - 4ac}$

$\sf\purple{ = 2^{2} - 4\times 8 \times (-3) }$

$\sf \purple{= 4 + 96}$

$\sf \purple{= 100 \gt 0}$

$\sf \therefore \green { Roots \: are \: real \:and \: distinct . }$

$\sf \red{ii) \ Compare \: given \: Quadratic \: equation}$

$\sf\orange{ -2x^2 + 3x + 2=0 \: with \: ax^{2}+bx+c=0 ,we \ get}$

$\sf \orange{a = -2 , \: b = 3 \:and \: c = 2}$

$\sf\orange{ Discreminant (D)}$

$\sf\orange{ = b^{2} - 4ac}$

$\sf\orange{ = 3^{2} - 4\times (-2) \times 2}$

$\sf \orange{= 9 + 16 }$

$\sf\orange{ = 25 \gt 0}$

$\sf \therefore \green { Roots \: are \: real \:and \: distinct . }$

$\sf\red{ iii) \ Compare \: given \: Quadratic \: equation}$

$\sf\blue{ 5x^2 -2 x - 10 =0 \: with \: ax^{2}+bx+c=0 ,we \ get }$

$\sf\blue{ a = 5 , \: b = -2 \:and \: c = -10}$

$\sf\blue{ Discreminant (D) }$

$\sf\blue{ = b^{2} - 4ac}$

$\sf \blue{= (-2)^{2} - 4\times 5 \times (-10)}$

$\sf \blue{= 4 + 200 }$

$\sf\blue{ = 204 \gt 0}$

$\sf \therefore \green { Roots \: are \: real \:and \: distinct . }$

$\sf\red{ iv) \ Given \: Quadratic \: equation :}$

$\sf\pink{ \dfrac{1}{2x-3} + \frac{1}{x-5} = 1 }$

$\sf\pink{ \to\:\: \dfrac{x-5+2x-3}{(2x-3)(x-5)} = 1 }$

$\sf \pink{\to\:\: \dfrac{ 3x-8 }{2x^{2} - 10x - 3x + 15} = 1 }$

$\sf\pink{\to\:\: \dfrac{ 3x-8 }{2x^{2} - 13x + 15} = 1 }$

$\sf \pink{\to\:\: 3x - 8 = 2x^{2} - 13x + 15}$

$\sf\pink{\to\:\: 0 = 2x^{2} - 13x + 15- 3x + 8 }$

$\sf\pink{ \to\:\: 2x^{2} - 16x + 23 = 0 }$

$\sf \pink{Compare \: given \: Quadratic \: equation}$

$\sf\pink{ 2x^2 -16x +23 =0 \: with \: ax^{2}+bx+c=0 ,we \ get }$

$\sf \pink{a = 2 , \: b = -16\:and \: c = 23}$

$\sf\pink{ Discreminant (D)}$

$\sf \pink{= b^{2} - 4ac}$

$\sf \pink{= (-16)^{2} - 4\times 2 \times 23}$

$\sf \pink{= 256 - 184}$

$\sf\pink{ = 72 \gt 0}$

$\sf \therefore \green { Roots \: are \: real \:and \: distinct . }$