Educational

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{x-2}{x-3}+\frac{x-4}{x-5}=\frac{10}{3}, \qquad x\ne 3,5 \] Solution Given: \[ \frac{x-2}{x-3}+\frac{x-4}{x-5}=\frac{10}{3} \] Multiplying both sides by \(3(x-3)(x-5)\): \[ 3(x-2)(x-5)+3(x-4)(x-3) =10(x-3)(x-5) \] \[ 3(x^2-7x+10)+3(x^2-7x+12) =10(x^2-8x+15) \] \[ 6x^2-42x+66 =10x^2-80x+150 \] \[ 4x^2-38x+84=0 \] \[ 2x^2-19x+42=0 \] Factorizing: \[ 2x^2-12x-7x+42=0 \] \[ 2x(x-6)-7(x-6)=0 \] \[ (x-6)(2x-7)=0 \] Therefore, \[ x-6=0 […]

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{x-4}{x-5}+\frac{x-6}{x-7}=\frac{10}{3}, \qquad x\ne 5,7 \] Solution Given: \[ \frac{x-4}{x-5}+\frac{x-6}{x-7}=\frac{10}{3} \] Cross-multiplying by \(3(x-5)(x-7)\): \[ 3(x-4)(x-7)+3(x-6)(x-5)=10(x-5)(x-7) \] \[ 3(x^2-11x+28)+3(x^2-11x+30) =10(x^2-12x+35) \] \[ 6x^2-66x+174 =10x^2-120x+350 \] \[ 4x^2-54x+176=0 \] \[ 2x^2-27x+88=0 \] Factorizing: \[ 2x^2-16x-11x+88=0 \] \[ 2x(x-8)-11(x-8)=0 \] \[ (x-8)(2x-11)=0 \] Therefore, \[ x-8=0 \quad \text{or} \quad

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{4}{x}-3=\frac{5}{2x+3}, \qquad x\ne 0,-\frac{3}{2} \] Solution Given: \[ \frac{4}{x}-3=\frac{5}{2x+3} \] Multiplying both sides by \(x(2x+3)\): \[ 4(2x+3)-3x(2x+3)=5x \] \[ 8x+12-6x^2-9x=5x \] \[ -6x^2-6x+12=0 \] \[ x^2+x-2=0 \] Factorizing: \[ x^2+2x-x-2=0 \] \[ x(x+2)-1(x+2)=0 \] \[ (x+2)(x-1)=0 \] Therefore, \[ x+2=0 \quad \text{or} \quad x-1=0 \] \[

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{x-1}{2x+1}+\frac{2x+1}{x-1}=\frac{5}{2}, \qquad x\ne -\frac12,\;1 \] Solution Given: \[ \frac{x-1}{2x+1}+\frac{2x+1}{x-1}=\frac{5}{2} \] Taking LCM on the left side: \[ \frac{(x-1)^2+(2x+1)^2}{(2x+1)(x-1)} =\frac{5}{2} \] Cross-multiplying: \[ 2\Big[(x-1)^2+(2x+1)^2\Big] =5(2x+1)(x-1) \] \[ 2\Big[(x^2-2x+1)+(4x^2+4x+1)\Big] =5(2x^2-x-1) \] \[ 10x^2+4x+4 =10x^2-5x-5 \] \[ 9x+9=0 \] \[ x+1=0 \] \[ x=-1 \] Since \(x=-1\) does not

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{x+1}{x-1}-\frac{x-1}{x+1}=\frac{5}{6}, \qquad x\ne 1,-1 \] Solution Given: \[ \frac{x+1}{x-1}-\frac{x-1}{x+1}=\frac{5}{6} \] Taking LCM on the left side: \[ \frac{(x+1)^2-(x-1)^2}{(x-1)(x+1)} =\frac{5}{6} \] Using the identity \(a^2-b^2=(a-b)(a+b)\): \[ \frac{\big[(x+1)-(x-1)\big]\big[(x+1)+(x-1)\big]} {x^2-1} =\frac{5}{6} \] \[ \frac{(2)(2x)}{x^2-1} =\frac{5}{6} \] \[ \frac{4x}{x^2-1} =\frac{5}{6} \] Cross-multiplying: \[ 24x=5(x^2-1) \] \[ 24x=5x^2-5 \] \[ 5x^2-24x-5=0

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{1}{x-2}+\frac{2}{x-1}=\frac{6}{x}, \qquad x\ne 0,1,2 \] Solution Given: \[ \frac{1}{x-2}+\frac{2}{x-1}=\frac{6}{x} \] Multiplying both sides by \(x(x-1)(x-2)\): \[ x(x-1)+2x(x-2)=6(x-1)(x-2) \] \[ x^2-x+2x^2-4x=6(x^2-3x+2) \] \[ 3x^2-5x=6x^2-18x+12 \] \[ 3x^2-13x+12=0 \] Factorizing: \[ 3x^2-9x-4x+12=0 \] \[ 3x(x-3)-4(x-3)=0 \] \[ (x-3)(3x-4)=0 \] Therefore, \[ x-3=0 \quad \text{or} \quad 3x-4=0 \] \[

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{x-3}{x+3}-\frac{x+3}{x-3}=\frac{48}{7}, \qquad x\ne 3,-3 \] Solution Given: \[ \frac{x-3}{x+3}-\frac{x+3}{x-3}=\frac{48}{7} \] Taking LCM on the left side: \[ \frac{(x-3)^2-(x+3)^2}{(x+3)(x-3)}=\frac{48}{7} \] Using \(a^2-b^2=(a-b)(a+b)\): \[ \frac{\big[(x-3)-(x+3)\big]\big[(x-3)+(x+3)\big]}{x^2-9} =\frac{48}{7} \] \[ \frac{(-6)(2x)}{x^2-9} =\frac{48}{7} \] \[ \frac{-12x}{x^2-9} =\frac{48}{7} \] Cross-multiplying: \[ -84x=48(x^2-9) \] \[ -84x=48x^2-432 \] \[ 48x^2+84x-432=0 \] Dividing by 12:

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{x+3}{x-2}-\frac{1-x}{x}=\frac{17}{4}, \qquad x\ne 0,2 \] Solution Given: \[ \frac{x+3}{x-2}-\frac{1-x}{x}=\frac{17}{4} \] Since \(1-x=-(x-1)\), we have \[ \frac{x+3}{x-2}+\frac{x-1}{x}=\frac{17}{4} \] Multiplying both sides by \(4x(x-2)\): \[ 4x(x+3)+4(x-1)(x-2)=17x(x-2) \] \[ 4x^2+12x+4(x^2-3x+2)=17x^2-34x \] \[ 8x^2+8+17x^2-34x \] \[ 8x^2+8=17x^2-34x \] \[ 9x^2-34x-8=0 \] Factorizing: \[ 9x^2-36x+2x-8=0 \] \[ 9x(x-4)+2(x-4)=0 \] \[ (9x+2)(x-4)=0

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{2x}{x-4}+\frac{2x-5}{x-3}=\frac{25}{3}, \qquad x\ne 3,4 \] Solution Given: \[ \frac{2x}{x-4}+\frac{2x-5}{x-3}=\frac{25}{3} \] Multiplying both sides by \(3(x-4)(x-3)\), we get: \[ 6x(x-3)+3(2x-5)(x-4)=25(x-4)(x-3) \] \[ 6x^2-18x+6x^2-39x+60=25x^2-175x+300 \] \[ 12x^2-57x+60=25x^2-175x+300 \] \[ 13x^2-118x+240=0 \] Factorizing: \[ 13x^2-78x-40x+240=0 \] \[ 13x(x-6)-40(x-6)=0 \] \[ (13x-40)(x-6)=0 \] Therefore, \[ 13x-40=0 \quad \text{or} \quad x-6=0

Solve the Following Quadratic Equation by Factorization Question: \[ \frac{x+3}{x+2}=\frac{3x-7}{2x-3}, \qquad x\ne -2,\frac{3}{2} \] Solution Given: \[ \frac{x+3}{x+2}=\frac{3x-7}{2x-3} \] Cross-multiplying, we get: \[ (x+3)(2x-3)=(x+2)(3x-7) \] Expanding both sides: \[ 2x^2+3x-9=3x^2-x-14 \] Bringing all terms to one side: \[ x^2-4x-5=0 \] Factorizing: \[ x^2-5x+x-5=0 \] \[ x(x-5)+1(x-5)=0 \] \[ (x-5)(x+1)=0 \] Therefore, \[ x-5=0 \quad \text{or}