please solve this problem , It's argent
Answers
We have,
Can be written as,
Making a Quadratic Equation,
We have,
Solving the Quadratic Equation by Quadratic Formula,
As we know,
Here,
Substituting the values and Finding the value of x,
Therefore,
- The values of x is 3 and -4.
- The values of x is 3 and -4.
Step-by-step explanation:
Question:
\begin{gathered}\tt (x) \: \frac{x}{x + 1} + \frac{x + 1}{x} = 2 \frac{1}{12} ,x ≠0, - 1 \\ \end{gathered}
(x)
x+1
x
+
x
x+1
=2
12
1
,x
=0,−1
\begin{gathered} \large \underline{ \underline{ \text{Solution:}}} \\ \end{gathered}
Solution:
We have,
\begin{gathered} \frac{x}{x + 1} + \frac{x + 1}{x} = 2 \frac{1}{12} \\ \end{gathered}
x+1
x
+
x
x+1
=2
12
1
Can be written as,
\begin{gathered} \frac{x}{x + 1} + \frac{x + 1}{x} = \frac{25}{12} \\ \end{gathered}
x+1
x
+
x
x+1
=
12
25
Making a Quadratic Equation,
\begin{gathered}\implies \frac{x}{x + 1} + \frac{x + 1}{x} = \frac{25}{12} \\ \\ \implies \frac{ {x}^{2} + (x + 1)(x + 1)}{x(x + 1)} = \frac{25}{12} \\ \\ \implies \frac{ {x}^{2} + {x}^{2} + 2x + 1 }{ {x}^{2} + x} = \frac{25}{12} \\ \\ \implies \frac{2 {x}^{2} + 2x + 1 }{ {x}^{2} + x} = \frac{25}{12} \\ \\ \implies 12(2 {x}^{2} + 2x + 1) = 25( {x}^{2} + x) \\ \\\implies 24 {x}^{2} + 24x + 12 = 25 {x}^{2} + 25x \\ \\ \implies 25 {x}^{2} + 25x - 24 {x}^{2} - 24x - 12 = 0 \\ \\ \implies {x}^{2} + x - 12 = 0 \\ \end{gathered}
⟹
x+1
x
+
x
x+1
=
12
25
⟹
x(x+1)
x
2
+(x+1)(x+1)
=
12
25
⟹
x
2
+x
x
2
+x
2
+2x+1
=
12
25
⟹
x
2
+x
2x
2
+2x+1
=
12
25
⟹12(2x
2
+2x+1)=25(x
2
+x)
⟹24x
2
+24x+12=25x
2
+25x
⟹25x
2
+25x−24x
2
−24x−12=0
⟹x
2
+x−12=0
We have,
\begin{gathered}{x}^{2} + x - 12 = 0 \\ \end{gathered}
x
2
+x−12=0
Solving the Quadratic Equation by Quadratic Formula,
As we know,
\begin{gathered}\boxed{x = \frac{ - b± \sqrt{ {b}^{2} - 4ac } }{2a} } \\ \end{gathered}
x=
2a
−b±
b
2
−4ac
Here,
\begin{gathered}a = 1 , \: b= 1 \: \text{and} \: c = -12 \\ \end{gathered}
a=1,b=1andc=−12
Substituting the values and Finding the value of x,
\begin{gathered}\implies x = \frac{ - 1± \sqrt{ {(1)}^{2} - 4(1)( - 12) } }{2(1)} \\ \\ \implies x = \frac{ - 1± \sqrt{1 + 48} }{2} \\ \\ \implies x = \frac{ - 1± \sqrt{49} }{2} \\ \\ \implies x = \frac{ - 1± 7}{2} \\ \\ \implies x = \frac{ - 1 + 7}{2} \: \text{and} \: \frac{ - 1 - 7}{2} \\ \\ \implies x = \frac{6}{2} \: \text{and} \: \frac{ -8}{2} \\ \\ \implies x = 3 \: \text{and} \: - 4 \\ \end{gathered}
⟹x=
2(1)
−1±
(1)
2
−4(1)(−12)
⟹x=
2
−1±
1+48
⟹x=
2
−1±
49
⟹x=
2
−1±7
⟹x=
2
−1+7
and
2
−1−7
⟹x=
2
6
and
2
−8
⟹x=3and−4
Therefore,
The values of x is 3 and -4.
\begin{gathered} \\ \large \underline{ \underline{ \text{Required Answer:}}} \\ \end{gathered}
Required Answer:
The values of x is 3 and -4.