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What is the number of integral solutions of the equation 2x2 – 3x – 2 = 0?
- a)0
- b)1
- c)2
- d)3
- e)4
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
What is the number of integral solutions of the equation 2x2 – 3...
- Using the quadratic formula:
- (3 ± √25) /4
- The two roots are:
- 8 /4 = 2
- -2 /4 = -1⁄2
- Since only x = 2 is an integer, the number of integral solutions is: 1
Most Upvoted Answer
What is the number of integral solutions of the equation 2x2 – 3...
Understanding the Equation
The given equation is 2x² - 3x - 2 = 0. To find the integral solutions, we will use the quadratic formula:
Quadratic Formula
The general form of a quadratic equation is ax² + bx + c = 0, where:
- a = 2
- b = -3
- c = -2
The quadratic formula states that the solutions for x can be found using:
x = [-b ± √(b² - 4ac)] / 2a
Calculating the Discriminant
First, we will calculate the discriminant (D):
- D = b² - 4ac
- D = (-3)² - 4(2)(-2)
- D = 9 + 16
- D = 25
Since the discriminant is a perfect square, there will be two real solutions.
Finding the Solutions
Now substituting the values into the quadratic formula:
x = [3 ± √25] / 4
x = [3 ± 5] / 4
This gives us two potential solutions:
1. x = (3 + 5) / 4 = 8 / 4 = 2
2. x = (3 - 5) / 4 = -2 / 4 = -0.5
Identifying Integral Solutions
Among these solutions, we need to identify the integral ones:
- The first solution is x = 2 (an integer).
- The second solution is x = -0.5 (not an integer).
Conclusion
Therefore, there is only one integral solution to the equation 2x² - 3x - 2 = 0.
The correct answer is option B: 1.
The given equation is 2x² - 3x - 2 = 0. To find the integral solutions, we will use the quadratic formula:
Quadratic Formula
The general form of a quadratic equation is ax² + bx + c = 0, where:
- a = 2
- b = -3
- c = -2
The quadratic formula states that the solutions for x can be found using:
x = [-b ± √(b² - 4ac)] / 2a
Calculating the Discriminant
First, we will calculate the discriminant (D):
- D = b² - 4ac
- D = (-3)² - 4(2)(-2)
- D = 9 + 16
- D = 25
Since the discriminant is a perfect square, there will be two real solutions.
Finding the Solutions
Now substituting the values into the quadratic formula:
x = [3 ± √25] / 4
x = [3 ± 5] / 4
This gives us two potential solutions:
1. x = (3 + 5) / 4 = 8 / 4 = 2
2. x = (3 - 5) / 4 = -2 / 4 = -0.5
Identifying Integral Solutions
Among these solutions, we need to identify the integral ones:
- The first solution is x = 2 (an integer).
- The second solution is x = -0.5 (not an integer).
Conclusion
Therefore, there is only one integral solution to the equation 2x² - 3x - 2 = 0.
The correct answer is option B: 1.
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