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If twice the reciprocal of x is equal to the sum of x and 1, how many values of x are possible?
- a)None
- b)One
- c)Two
- d)Three
- e)An infinite number
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If twice the reciprocal of x is equal to the sum of x and 1, how many ...
Step 1: Set up the equation based on the description
Step 2 & 3: Finding required values and calculating the final answer
This is a quadratic equation, so x has either one or two possible answers (Every quadratic equation has two roots but variable x may have some constraints imposed on it, which make one of the roots an impossible value to have for x)
The constraint on x in the given equation is that x cannot be equal to zero (since x is in the denominator).
To find out how many values of x are possible, we set up the equation to find the roots.
First, multiply both sides by x to get rid of the fraction:
2=x2+x
Next, set the equation equal to zero:
x2+x−2=0
Finally, factor out the expressions:
(x+2)(x−1)=0
x has two possible values: -2 and 1,
Answer: Option (C)
Most Upvoted Answer
If twice the reciprocal of x is equal to the sum of x and 1, how many ...
Problem Analysis:
We are given that twice the reciprocal of x is equal to the sum of x and 1. Let's break down the statement:
- The reciprocal of x is 1/x.
- Twice the reciprocal of x is 2/x.
- The sum of x and 1 is x + 1.
So, we can write the given equation as 2/x = x + 1.
Solving the Equation:
To solve the equation, we can multiply both sides by x to eliminate the fraction:
2 = x(x + 1).
Expanding the equation:
2 = x^2 + x.
Rearranging the equation to standard quadratic form:
x^2 + x - 2 = 0.
Now, we need to solve this quadratic equation to find the possible values of x.
Factoring the Quadratic Equation:
To factor the quadratic equation, we need to find two numbers whose product is -2 and whose sum is 1.
The numbers that satisfy these conditions are 2 and -1:
x^2 + 2x - x - 2 = 0.
x(x + 2) - 1(x + 2) = 0.
(x - 1)(x + 2) = 0.
Setting each factor equal to zero:
x - 1 = 0 or x + 2 = 0.
Solving for x:
x = 1 or x = -2.
Conclusion:
We have found two values of x that satisfy the given equation: x = 1 and x = -2. Therefore, the correct answer is option C: Two.
We are given that twice the reciprocal of x is equal to the sum of x and 1. Let's break down the statement:
- The reciprocal of x is 1/x.
- Twice the reciprocal of x is 2/x.
- The sum of x and 1 is x + 1.
So, we can write the given equation as 2/x = x + 1.
Solving the Equation:
To solve the equation, we can multiply both sides by x to eliminate the fraction:
2 = x(x + 1).
Expanding the equation:
2 = x^2 + x.
Rearranging the equation to standard quadratic form:
x^2 + x - 2 = 0.
Now, we need to solve this quadratic equation to find the possible values of x.
Factoring the Quadratic Equation:
To factor the quadratic equation, we need to find two numbers whose product is -2 and whose sum is 1.
The numbers that satisfy these conditions are 2 and -1:
x^2 + 2x - x - 2 = 0.
x(x + 2) - 1(x + 2) = 0.
(x - 1)(x + 2) = 0.
Setting each factor equal to zero:
x - 1 = 0 or x + 2 = 0.
Solving for x:
x = 1 or x = -2.
Conclusion:
We have found two values of x that satisfy the given equation: x = 1 and x = -2. Therefore, the correct answer is option C: Two.
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