Question

$\sf \dfrac {2x+5}{3x-3}=\dfrac {5x+8}{4x-5}$

$\ltexts {Solve\:it}$​

Answer 1

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

$\sf= > \dfrac{2x + 5}{3x - 3} = \dfrac{5x + 8}{4x - 5}$

Now,cross multiply to both sides:

=>(4x-5)(2x+5)=(3x-3)(5x+8)

=>4x(2x+5)-5(2x+5)=3x(5x+8)-3(5x+8)

=>8x²+20x-10x-25=15x²+24x-15x-24

=>8x²+10x-25=15x²+9x-24

Arranging like terms together:-

=>15x²-8x²+9x-10x-24+25

=>7x²-x+1=0

it is in the form of a quadratic equation so we use quadratic formula for finding the roots.

x= -b ±√b²-4ac / 2a

=>7x²-x+1=0

a=7 ,b = -1 & c= 1

=>x= -(-1) ± √(-1)²-4(7)(1) / 2(7)

=>x= 1± √1-28 / 14

=>x= 1 ±√-27 / 14

Here,the discriminant b²-4ac is < 0 .

So,there are two complex roots.

Here, i ( iota ) is used when there is negative value inside root.

=>x= 1±3√3i / 14

=>x= 1/14 ± 3√3i /14

x= 0.0714286 + 0.371154i

x= 0.0714286 - 0.371154i

Answer 2

⠀⠀⠀⠀⠀⠀${\rm{\pink{\underline{\underline{★Answer★}}}}}$

${\rm{\red{\underline{\underline{Given\: Equation : }}}}}$

$\rm\dfrac {2x+5}{3x-3}=\dfrac {5x+8}{4x-5}$

${\rm{\purple{\underline{\underline{Solution}}}}}$

$\rm\dfrac {2x+5}{3x-3}=\dfrac {5x+8}{4x-5}$

Cross multiplying the terms, we get :-

$\longrightarrow\rm4x - 5(2x + 5) = 3x - 3(5x + 8)$

$\longrightarrow\rm 8x {}^{2} + 20x - 10 {x}^{} - 25 = 15x {}^{2} + 24x - 15x - 24$

$\longrightarrow\rm 8x {}^{2} + 10x - 25 = 15x {}^{2} + 9x - 24$

Taking all the terms to the L.H.S, we get :-

$\longrightarrow\rm 8x {}^{2} - 15x {}^{2} + 10x - 9x - 25 + 24 = 0$

$\longrightarrow\rm - 7x {}^{2} + x - 1 = 0$

Multiplying all the terms by ( - ), we get :-

$\longrightarrow\rm - ( - 7x {}^{2} + x - 1) = 0$

$\longrightarrow\rm 7x {}^{2} - x + 1 = 0$

By using the quadratic equation, we get :-

$\longrightarrow\rm x = \frac{- b± \sqrt{} (b {}^{2} - 4ac)}{2a}$

Here,

a = 7

b = 1

c = 1

Now, substituting the values,

$\longrightarrow\rm x = \frac{ - 1± \sqrt{} [( 1) {}^{2} - 4 \times 7 \times 1]}{2(7)}7$

$\longrightarrow\rm x = \frac{ - 1 ±\sqrt{1 - 28} }{14}$

$\longrightarrow\rm x = \frac{ - 1 \: ±\sqrt{} - 27}{14}$

No real solutions because the discriminant is negative.

RESULT:- NO SOLUTION