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If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0?
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If one root of the quadratic equation is 2, find the value of p. Also,...
Given Quadratic Equation
The quadratic equation we are examining is:
3x² - px - 2 = 0
Finding the Value of p
Since one root is given as 2, we can substitute x = 2 into the equation:
3(2)² - p(2) - 2 = 0
Calculating the left side:
- 3(4) - 2p - 2 = 0
- 12 - 2p - 2 = 0
- 10 - 2p = 0
Now, solve for p:
- 2p = 10
- p = 5
Finding the Other Root
To find the other root (let's call it r), we can utilize Vieta's formulas, which states that the sum of the roots (2 + r) is equal to the negation of the coefficient of x divided by the coefficient of x²:
- Sum of roots: 2 + r = p/3
- Substituting p = 5: 2 + r = 5/3
- Rearranging: r = 5/3 - 2 = 5/3 - 6/3 = -1
Nature of Roots
To analyze the nature of the roots, we can calculate the discriminant (D):
D = b² - 4ac
For our equation:
- a = 3, b = -p = -5, c = -2
Plugging values into the formula:
D = (-5)² - 4(3)(-2)
D = 25 + 24
D = 49
Since D > 0, the quadratic equation has:
- Two distinct real roots.
Summary
- Value of p: 5
- Other root: -1
- Nature of roots: Two distinct real roots.
The quadratic equation we are examining is:
3x² - px - 2 = 0
Finding the Value of p
Since one root is given as 2, we can substitute x = 2 into the equation:
3(2)² - p(2) - 2 = 0
Calculating the left side:
- 3(4) - 2p - 2 = 0
- 12 - 2p - 2 = 0
- 10 - 2p = 0
Now, solve for p:
- 2p = 10
- p = 5
Finding the Other Root
To find the other root (let's call it r), we can utilize Vieta's formulas, which states that the sum of the roots (2 + r) is equal to the negation of the coefficient of x divided by the coefficient of x²:
- Sum of roots: 2 + r = p/3
- Substituting p = 5: 2 + r = 5/3
- Rearranging: r = 5/3 - 2 = 5/3 - 6/3 = -1
Nature of Roots
To analyze the nature of the roots, we can calculate the discriminant (D):
D = b² - 4ac
For our equation:
- a = 3, b = -p = -5, c = -2
Plugging values into the formula:
D = (-5)² - 4(3)(-2)
D = 25 + 24
D = 49
Since D > 0, the quadratic equation has:
- Two distinct real roots.
Summary
- Value of p: 5
- Other root: -1
- Nature of roots: Two distinct real roots.
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If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0?
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If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0?.
If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If one root of the quadratic equation is 2, find the value of p. Also, find the other root and nature of roots 3x2 - px - 2 = 0?.
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