Question
Solve the following quadratic equations by factorization:
$\frac{1}{\text{x}-3}+\frac{2}{\text{x}-2}=\frac{8}{\text{x}},$ $\text{x}\neq0, 2, 3$
$\frac{1}{\text{x}-3}+\frac{2}{\text{x}-2}=\frac{8}{\text{x}},$ $\text{x}\neq0, 2, 3$
✓
Answer
$\frac{1}{\text{x}-3}+\frac{2}{\text{x}-2}=\frac{8}{\text{x}}$
$\Rightarrow\frac{(\text{x}-2)+2(\text{x}-3)}{(\text{x}-3)(\text{x}-2)}=\frac{8}{\text{x}}$
$\Rightarrow\frac{\text{x}-2+2\text{x}-6}{\text{x}^2-2\text{x}-3\text{x}+6}=\frac{8}{\text{x}}$
$\Rightarrow\frac{3\text{x}-8}{\text{x}^2-5\text{x}+6}=\frac{8}{\text{x}}$
$\Rightarrow x(3x - 8) = 8(x^2 - 5x + 6)$
$\Rightarrow 3x^2 - 8x = 8x^2 - 40x + 48$
$\Rightarrow 5x^2 - 32x + 48 = 0$
$\Rightarrow 5x^2 - 20x - 12x + 48 = 0$
$\Rightarrow 5x(x - 4) - 12(x - 4) = 0$
$\Rightarrow (5x - 12)(x - 4) = 0$
$\Rightarrow 5x - 12 = 0$ or $x - 4 = 0$
$\Rightarrow\text{x}=\frac{12}{5}$ or $x = 4$
Hence, the factors 4 and $\frac{12}{5}$
$\Rightarrow\frac{(\text{x}-2)+2(\text{x}-3)}{(\text{x}-3)(\text{x}-2)}=\frac{8}{\text{x}}$
$\Rightarrow\frac{\text{x}-2+2\text{x}-6}{\text{x}^2-2\text{x}-3\text{x}+6}=\frac{8}{\text{x}}$
$\Rightarrow\frac{3\text{x}-8}{\text{x}^2-5\text{x}+6}=\frac{8}{\text{x}}$
$\Rightarrow x(3x - 8) = 8(x^2 - 5x + 6)$
$\Rightarrow 3x^2 - 8x = 8x^2 - 40x + 48$
$\Rightarrow 5x^2 - 32x + 48 = 0$
$\Rightarrow 5x^2 - 20x - 12x + 48 = 0$
$\Rightarrow 5x(x - 4) - 12(x - 4) = 0$
$\Rightarrow (5x - 12)(x - 4) = 0$
$\Rightarrow 5x - 12 = 0$ or $x - 4 = 0$
$\Rightarrow\text{x}=\frac{12}{5}$ or $x = 4$
Hence, the factors 4 and $\frac{12}{5}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In figure 3.103, seg AD $\perp$ side BC, seg BE $\perp$ side AC, seg CF $\perp$ side AB. Point O is the orthocentre. Prove that , point O is the incentre of∆ DEF.
→If $\cos\theta=\frac{7}{25},$ find the value of the all T-ratios of $\theta.$
→Solve for x and y:
$\frac{5}{\text{x}}+\text{6y}=13,$
$\frac{3}{\text{x}}+\text{4y}=7\ (\text{x}\neq0).$
→$\frac{5}{\text{x}}+\text{6y}=13,$
$\frac{3}{\text{x}}+\text{4y}=7\ (\text{x}\neq0).$
M/S Shridhar Chemicals purchased washing powder for ₹ 10,000 taxable amount. They sold it to a shopkeeper for ₹ 12,000 taxable amount. The rate of GST is $18 \%$, then find the CGST and SGST to be paid by M/S Shridhar Chemicals.
→Nalinitai invested ₹ 6024 in the shares of FV ₹ 10 when the Market Value was ₹ 60. She sold all the shares at MV of ₹ 50 after taking 60% dividend. She paid 0.4% brokerage at each stage of transactions. What was the total gain or loss in this transaction ?
Show that the points A(3, 0), B(4, 5), C(-1, 4) and D(-2, -1) are the vertices of a rhombus. Find its area.
If $(m+1)^{\text {th }}$ term of an A.P is twice the $(n+1)^{\text {th }}$ term, prove that $(3 m+1)^{\text {th }}$ term is twice the $(m+n+1)^{\text {th }}$ term.
→How many numbers are there between $101$ and $999$, which are divisible by both $2$ and $5$?
→Pranali and Prasad started walking to the East and to the North respectively, from the same point and at the same speed. After 2 hours distance between them was 15$\sqrt 2$ km. Find their speed per hour.
→$x+2 y+4=0 ; 3 x=-4 y-16$
→