<u>Answer:</u>
a. For equation
value of x are 1 and -6.
b. For equation
value of x are 3 and -8
c. For equation
value of x are -1 and 1.
<u>Solution:</u>
a)Given equation is ![x+5=\frac{6}{x}](https://tex.z-dn.net/?f=x%2B5%3D%5Cfrac%7B6%7D%7Bx%7D)
On cross multiplying we get,
x(x+5)=6
![x^{2}+5 x-6=0](https://tex.z-dn.net/?f=x%5E%7B2%7D%2B5%20x-6%3D0)
On splitting the middle term we get,
![\begin{array}{l}{=x^{2}+6 x-x-6=0} \\ {=x(x+6)-1(x+6)=0} \\ {=(x-1)(x+6)=0}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B%3Dx%5E%7B2%7D%2B6%20x-x-6%3D0%7D%20%5C%5C%20%7B%3Dx%28x%2B6%29-1%28x%2B6%29%3D0%7D%20%5C%5C%20%7B%3D%28x-1%29%28x%2B6%29%3D0%7D%5Cend%7Barray%7D)
When x - 1 = 0 , x = 1
When x + 6 = 0 , x = -6
So two values of x which satisfies the given equation are 1 and -6.
b)Given equation is ![x=\frac{24}{x}-5](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B24%7D%7Bx%7D-5)
![\begin{array}{l}{=x \times x=24-5 x} \\ {=x^{2}+5 x-24=0}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B%3Dx%20%5Ctimes%20x%3D24-5%20x%7D%20%5C%5C%20%7B%3Dx%5E%7B2%7D%2B5%20x-24%3D0%7D%5Cend%7Barray%7D)
On splitting the middle term we get
![\begin{array}{l}{=x^{2}+8 x-3 x-24=0} \\ {=x(x+8)-3(x+8)=0} \\ {=(x-3)(x+8)=0}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B%3Dx%5E%7B2%7D%2B8%20x-3%20x-24%3D0%7D%20%5C%5C%20%7B%3Dx%28x%2B8%29-3%28x%2B8%29%3D0%7D%20%5C%5C%20%7B%3D%28x-3%29%28x%2B8%29%3D0%7D%5Cend%7Barray%7D)
When x - 3 = 0 , x = 3
When x + 8 = 0 , x = -8
So two values of x which satisfies the given equation are 3 and -8.
c) Given equation is ![x=\frac{1}{x}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B1%7D%7Bx%7D)
![\begin{array}{l}{=x \times x=1} \\ {=x^{2}-1=0} \\ {=x^{2}-1^{2}=0}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B%3Dx%20%5Ctimes%20x%3D1%7D%20%5C%5C%20%7B%3Dx%5E%7B2%7D-1%3D0%7D%20%5C%5C%20%7B%3Dx%5E%7B2%7D-1%5E%7B2%7D%3D0%7D%5Cend%7Barray%7D)
Using algebraic identity
we get
=(x-1)(x+1)=0
When x - 1 = 0 , x =1
When x + 1 = 0 , x = -1
So two values of x which satisfies the given equation are 1 and -1.