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Mathematics

Solve the following equation by factorisation:

3x−22x−3=3x−8x+4\dfrac{3x - 2}{2x - 3} = \dfrac{3x - 8}{x + 4}

Quadratic Equations

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Answer

⇒3x−22x−3=3x−8x+4⇒(3x−2)(x+4)=(3x−8)(2x−3)⇒3x2+12x−2x−8=6x2−9x−16x+24⇒3x2+10x−8=6x2−25x+24⇒6x2−3x2−25x−10x+24+8=0⇒3x2−35x+32=0⇒3x2−32x−3x+32=0⇒x(3x−32)−1(3x−32)=0⇒(x−1)(3x−32)=0⇒x−1=0 or 3x−32=0⇒x=1 or x=323.⇒x=1 or x=1023.\phantom{\Rightarrow} \dfrac{3x - 2}{2x - 3} = \dfrac{3x - 8}{x + 4} \\[1em] \Rightarrow (3x - 2)(x + 4) = (3x - 8)(2x - 3) \\[1em] \Rightarrow 3x^2 + 12x - 2x - 8 = 6x^2 - 9x - 16x + 24 \\[1em] \Rightarrow 3x^2 + 10x - 8 = 6x^2 - 25x + 24 \\[1em] \Rightarrow 6x^2 - 3x^2 - 25x - 10x + 24 + 8 = 0 \\[1em] \Rightarrow 3x^2 - 35x + 32 = 0 \\[1em] \Rightarrow 3x^2 - 32x - 3x + 32 = 0 \\[1em] \Rightarrow x(3x - 32) - 1(3x - 32) = 0 \\[1em] \Rightarrow (x - 1)(3x - 32) = 0 \\[1em] \Rightarrow x - 1 = 0 \text{ or } 3x - 32 = 0 \\[1em] \Rightarrow x = 1 \text{ or } x = \dfrac{32}{3}. \\[1em] \Rightarrow x = 1 \text{ or } x = 10\dfrac{2}{3}.

Hence, x = 1 or x = 102310\dfrac{2}{3}.

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