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The number of real-valued solutions of the equation 2x+ 2-x = 2 - (x - 2)2 is:
- a)1
- b)2
- c)infinite
- d)0
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The number of real-valued solutions of the equation 2x+ 2-x= 2 - (x - ...
We notice that the minimum value of the term in the LHS will be greater than or equal to 2 {at x=0; LHS = 2}. However, the term in the RHS is less than or equal to 2 {at x=2; RHS = 2}. The values of x at which both the sides become 2 are distinct; hence, there are zero real-valued solutions to the above equation.
Most Upvoted Answer
The number of real-valued solutions of the equation 2x+ 2-x= 2 - (x - ...
Understanding the Equation
To solve the equation 2x + 2 - x = 2 - (x - 2)², we first simplify both sides.
Left Side Simplification
- The left side is: 2x + 2 - x = x + 2.
Right Side Simplification
- The right side expands as follows:
- (x - 2)² = x² - 4x + 4
- Therefore, 2 - (x - 2)² = 2 - (x² - 4x + 4) = -x² + 4x - 2.
So, the equation simplifies to:
x + 2 = -x² + 4x - 2.
Rearranging the Equation
Now, we bring all terms to one side:
x² - 3x + 4 = 0.
Discriminant Calculation
Next, we check the discriminant (D) of this quadratic equation:
- D = b² - 4ac
- Here, a = 1, b = -3, c = 4.
- D = (-3)² - 4(1)(4) = 9 - 16 = -7.
Conclusion on Real Solutions
Since the discriminant is negative (D < 0),="" the="" quadratic="" equation="" has="" no="" real="" solutions.="" thus,="" the="" number="" of="" real-valued="" solutions="" for="" the="" given="" equation="" is:="" />Final Answer
- 0 (option D).
To solve the equation 2x + 2 - x = 2 - (x - 2)², we first simplify both sides.
Left Side Simplification
- The left side is: 2x + 2 - x = x + 2.
Right Side Simplification
- The right side expands as follows:
- (x - 2)² = x² - 4x + 4
- Therefore, 2 - (x - 2)² = 2 - (x² - 4x + 4) = -x² + 4x - 2.
So, the equation simplifies to:
x + 2 = -x² + 4x - 2.
Rearranging the Equation
Now, we bring all terms to one side:
x² - 3x + 4 = 0.
Discriminant Calculation
Next, we check the discriminant (D) of this quadratic equation:
- D = b² - 4ac
- Here, a = 1, b = -3, c = 4.
- D = (-3)² - 4(1)(4) = 9 - 16 = -7.
Conclusion on Real Solutions
Since the discriminant is negative (D < 0),="" the="" quadratic="" equation="" has="" no="" real="" solutions.="" thus,="" the="" number="" of="" real-valued="" solutions="" for="" the="" given="" equation="" is:="" />Final Answer
- 0 (option D).
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