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One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent?
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One subtracting any two quadratic equations, after making their coeffi...
Condition for both roots common:
Let α, β be the common roots of the quadratic equations a1x^2 + b1x + c1 = 0 and a2x^2 + b2x + c2 = 0. Then
α + β = -b1/a1, αβ = c1/a1 and α + β = -b2/a2, αβ = c2/a2
Therefore, -b/a1 = - b2/a2 and c1/a1 = c2/a2
⇒ a1/a2 = b1/b2 and a1/a2 = c1/c2
⇒ a1/a2 = b1/b2 = c1/c2
This is the required condition.
Solved examples to find the conditions for one common root or both common roots of quadratic equations:
1. If the equations x^2 + px + q = 0 and x^2 + px + q = 0 have a common root and p ≠ q, then prove that p + q + 1 = 0.
Solution:
Let α be the common root of x^2 + px + q = 0 and x^2 + px + q = 0.
Then,
α^2 + pα + q = 0 and α^2 + pα + q = 0.
Subtracting second form the first,
α(p - q) + (q - p) = 0
⇒ α(p - q) - (p - q) = 0
⇒ (p - q)(α - 1) = 0
⇒ (α - 1) = 0, [p - q ≠0, since, p ≠ q]
⇒ α = 1
Therefore, from the equation α^2 + pα + q = 0 we get,
1^2 + p(1) + q = 0
⇒ 1 + p + q = 0
⇒ p + q + 1 = 0
Proved
Community Answer
One subtracting any two quadratic equations, after making their coeffi...
Explanation:
Subtracting Quadratic Equations:
- When we subtract any two quadratic equations after making their coefficient of x^2 the same, we are essentially finding the difference between the two equations.
- This process involves subtracting the corresponding coefficients of x^2, x, and the constant term in the equations.
Finding the Value of X:
- After subtracting the two quadratic equations, we simplify the resulting equation to get a new quadratic equation.
- By solving this new equation, we can find the value(s) of x that satisfy the given condition.
Representation of No Common Root:
- When the two original quadratic equations do not have a common root, it means that the solutions to each equation are different.
- In this case, the resulting equation obtained by subtraction represents the values of x for which the two original equations are not equal.
Interpretation:
- The value of x obtained from the subtraction of the two quadratic equations represents the points of difference between the two equations.
- These points indicate the specific values of x where the two equations diverge and have distinct solutions.
Conclusion:
- Subtracting quadratic equations and analyzing the resulting equation can provide valuable insights into the differences between the two equations and their solutions.
- Understanding the representation of no common root helps in interpreting the significance of the resulting value of x in the context of the original equations.
Subtracting Quadratic Equations:
- When we subtract any two quadratic equations after making their coefficient of x^2 the same, we are essentially finding the difference between the two equations.
- This process involves subtracting the corresponding coefficients of x^2, x, and the constant term in the equations.
Finding the Value of X:
- After subtracting the two quadratic equations, we simplify the resulting equation to get a new quadratic equation.
- By solving this new equation, we can find the value(s) of x that satisfy the given condition.
Representation of No Common Root:
- When the two original quadratic equations do not have a common root, it means that the solutions to each equation are different.
- In this case, the resulting equation obtained by subtraction represents the values of x for which the two original equations are not equal.
Interpretation:
- The value of x obtained from the subtraction of the two quadratic equations represents the points of difference between the two equations.
- These points indicate the specific values of x where the two equations diverge and have distinct solutions.
Conclusion:
- Subtracting quadratic equations and analyzing the resulting equation can provide valuable insights into the differences between the two equations and their solutions.
- Understanding the representation of no common root helps in interpreting the significance of the resulting value of x in the context of the original equations.
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One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent?
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One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent?.
One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for One subtracting any two quadratic equations, after making their coefficient of x^2 same, we get some value of X. If the equations do not have a common root, then what does it represent?.
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