Question
Solve the following quadratic equations by factorization:
$\frac{\text{a}}{\text{x}-\text{b}}+\frac{\text{b}}{\text{x}-\text{a}}=2$
$\frac{\text{a}}{\text{x}-\text{b}}+\frac{\text{b}}{\text{x}-\text{a}}=2$
✓
Answer
$\frac{\text{a}}{\text{x}-\text{b}}+\frac{\text{b}}{\text{x}-\text{a}}=2$
$\Rightarrow\frac{\text{a}(\text{x}-\text{a})+\text{b}(\text{x}-\text{b})}{(\text{x}-\text{a})(\text{x}-\text{b})}=2$
$\Rightarrow ax - a^2 + bx - b^2 = 2x^2 - 2ax - 2bx + 2ab$
$\Rightarrow 2x^2 - 2ax - ax - 2bx - bx + a^2 + b^2 + 2ab = 0$
$\Rightarrow 2x^2 - 3x(a + b) + (a + b)^2 = 0$
$\Rightarrow 2x^2 - 2x(a + b) - x(a + b) + (a + b)^2= 0$
$\Rightarrow 2x[x - (a + b)] - (a + b)[x - (a + b)] = 0$
$\Rightarrow [2x - (a + b)] [(x - a + b)] = 0$
$\Rightarrow\text{x}=\frac{\text{a}+\text{b}}{2}$ or x = a + b
$\Rightarrow\frac{\text{a}(\text{x}-\text{a})+\text{b}(\text{x}-\text{b})}{(\text{x}-\text{a})(\text{x}-\text{b})}=2$
$\Rightarrow ax - a^2 + bx - b^2 = 2x^2 - 2ax - 2bx + 2ab$
$\Rightarrow 2x^2 - 2ax - ax - 2bx - bx + a^2 + b^2 + 2ab = 0$
$\Rightarrow 2x^2 - 3x(a + b) + (a + b)^2 = 0$
$\Rightarrow 2x^2 - 2x(a + b) - x(a + b) + (a + b)^2= 0$
$\Rightarrow 2x[x - (a + b)] - (a + b)[x - (a + b)] = 0$
$\Rightarrow [2x - (a + b)] [(x - a + b)] = 0$
$\Rightarrow\text{x}=\frac{\text{a}+\text{b}}{2}$ or x = a + b
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
Draw a circle of radius 3cm. Take two points P and Q on one of its extended diameter each at a distance of 7cm from its centre. Draw tangents to the circle from these two points P and Q.
The difference of two numbers is $5$ and the difference of their reciprocals is $\frac{1}{10}$. Find the numbers.
→Places A and B are 160km apart on a highway. One car starts from A and another car from B at the same time. If they travel in the same direction, they meet in 8 hours. But, if they travel towards each other, they meet in 2 hours. Find the speed of each car.
A regular hexagon is inscribed in a circle. If the area of hexagon is $24\sqrt{3}\text{cm}^2,$ find the area of the circle. $\big(\text{Use }\pi=3.14\big)$
→In the following figure, there are three semicircles, A, B and C having diameter 3cm each, and another semicircle E having a circle D with diameter 4.5cm are shown. Calculate:
The area of the shaded region.
The cost of painting the shaded region at the rate of 25 paise per $cm^2$ , to the nearest rupee.
→The area of the shaded region.
The cost of painting the shaded region at the rate of 25 paise per $cm^2$ , to the nearest rupee.
At a point $A, 20$ metres above the level of water in a lake, the angle of elevation of a cloud is $30^{\circ}$. The angle of depression of the reflection of the cloud in the lake, at $A$ is $60^{\circ}$. Find the distance of the cloud from $A$.
→How many terms are there in the A.P. whose first and fifth term are -14 and 2 respectively and the sum if the terms is 40?
The sum $4^{\text {th }}$ and $8^{\text {th }}$ terms of an A.P. is 24 and the sum of $6^{\text {th }}$ and $10^{\text {th }}$ terms is 44 . Find the A.P.?
→Solve the following quadratic equations by factorization:
$\frac{1}{\text{x}-2}+\frac{2}{\text{x}-1}=\frac{6}{\text{x}},$ $\text{x}\neq0$
→$\frac{1}{\text{x}-2}+\frac{2}{\text{x}-1}=\frac{6}{\text{x}},$ $\text{x}\neq0$
Two tangents $TP$ and $TQ$ are drawn to a circle with centre $O$ from an external point $T$. Prove that $\angle PIQ =2 \angle OPQ$.
→