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The equations x2 +x +a=0 and x2 +ax +1=0 have a common real root
- a)for no value of a
- b)for exactly one value of a
- c)for exactly two values of a
- d)for exactly three values of a
Correct answer is option 'B'. Can you explain this answer?
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The equations x2 +x +a=0 and x2 +ax +1=0 have a common real roota)for ...
The Common Real Root of Two Quadratic Equations
To find the common real root of two quadratic equations, we need to equate the two equations and solve for the common value of x. Let's analyze the given equations and find the common real root.
Given Equations:
1) x^2 + x + a = 0
2) x^2 + ax + 1 = 0
Equating the Equations:
Let's equate the two equations to find the common root x:
x^2 + x + a = x^2 + ax + 1
Simplifying the Equation:
By rearranging the terms, we get:
x + a = ax + 1
Isolating the Variables:
To isolate x, we subtract ax from both sides of the equation:
x - ax + a = 1
Factoring Out x:
Next, we factor out x from the left side of the equation:
x(1 - a) + a = 1
Simplifying the Equation:
Now, we simplify the equation:
x(1 - a) = 1 - a
Dividing Both Sides by (1 - a):
To solve for x, we divide both sides of the equation by (1 - a):
x = (1 - a) / (1 - a)
Final Conclusion:
We can see that x is equal to 1 for all values of a except when a = 1. In this case, the denominator becomes 0, which is undefined. Therefore, for all other values of a, the common real root of the given equations is x = 1.
Answer:
The given equations have a common real root for exactly one value of a, which is option 'B'.
To find the common real root of two quadratic equations, we need to equate the two equations and solve for the common value of x. Let's analyze the given equations and find the common real root.
Given Equations:
1) x^2 + x + a = 0
2) x^2 + ax + 1 = 0
Equating the Equations:
Let's equate the two equations to find the common root x:
x^2 + x + a = x^2 + ax + 1
Simplifying the Equation:
By rearranging the terms, we get:
x + a = ax + 1
Isolating the Variables:
To isolate x, we subtract ax from both sides of the equation:
x - ax + a = 1
Factoring Out x:
Next, we factor out x from the left side of the equation:
x(1 - a) + a = 1
Simplifying the Equation:
Now, we simplify the equation:
x(1 - a) = 1 - a
Dividing Both Sides by (1 - a):
To solve for x, we divide both sides of the equation by (1 - a):
x = (1 - a) / (1 - a)
Final Conclusion:
We can see that x is equal to 1 for all values of a except when a = 1. In this case, the denominator becomes 0, which is undefined. Therefore, for all other values of a, the common real root of the given equations is x = 1.
Answer:
The given equations have a common real root for exactly one value of a, which is option 'B'.
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Community Answer
The equations x2 +x +a=0 and x2 +ax +1=0 have a common real roota)for ...
Condition for exactly one common root =>
(a1c2 - a2c1)² = (a1b2 - a2b1)(b1c2 - b2c1)
(a1c2 - a2c1)² = (a1b2 - a2b1)(b1c2 - b2c1)
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The equations x2 +x +a=0 and x2 +ax +1=0 have a common real roota)for no value of ab)for exactly one value of ac)for exactly two values of ad)for exactly three values of aCorrect answer is option 'B'. Can you explain this answer?
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