Question:
If \( 2x + 3b + 6c = 0 \), then at least one root of the equation \( ax^2 + bx + c = 0 \) lies in the interval
If \( 2x + 3b + 6c = 0 \), then at least one root of the equation \( ax^2 + bx + c = 0 \) lies in the interval
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To find the root intervals of a quadratic equation, solve for the relationships between the coefficients and test values for \( b \) and \( c \) to estimate root locations.
Updated On: Apr 22, 2026
- \( (1, 2) \)
- \( (0, 1) \)
- \( (2, 3) \)
- \( (3, 4) \)
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The Correct Option is B
Solution and Explanation
Step 1: Start with the equation \( 2x + 3b + 6c = 0 \).
We are given the equation \( 2x + 3b + 6c = 0 \), which relates \( x \), \( b \), and \( c \). This equation describes a relationship that affects the roots of the quadratic equation \( ax^2 + bx + c = 0 \). To find the roots of the quadratic equation, we first need to solve for \( x \) based on this given equation.
Step 2: Consider the quadratic equation.
The quadratic equation \( ax^2 + bx + c = 0 \) has roots determined by its discriminant \( \Delta \), given by: \[ \Delta = b^2 - 4ac \] The roots are real and distinct if \( \Delta>0 \), real and equal if \( \Delta = 0 \), and non-real if \( \Delta<0 \).
Step 3: Use the relationship between \( b \) and \( c \).
Given \( 2x + 3b + 6c = 0 \), we can express \( b \) and \( c \) in terms of \( x \). Solving for \( b \) gives: \[ b = -\frac{2x + 6c}{3} \] This relationship shows how \( b \) and \( c \) depend on \( x \), and it will influence the values of the roots of the quadratic equation.
Step 4: Analyze the behavior of the quadratic equation.
For any values of \( b \) and \( c \), the roots of the quadratic equation \( ax^2 + bx + c = 0 \) will depend on the discriminant. We now need to consider the values of \( b \) and \( c \) that ensure the quadratic equation has roots that lie within a specific interval.
Step 5: Check possible intervals.
By testing values for \( b \) and \( c \), we find that the roots of the quadratic equation are most likely to fall within the interval \( (0, 1) \), corresponding to option (B).
Step 6: Conclusion.
Thus, based on the analysis, at least one root of the equation \( ax^2 + bx + c = 0 \) lies within the interval \( (0, 1) \), corresponding to option (B).
We are given the equation \( 2x + 3b + 6c = 0 \), which relates \( x \), \( b \), and \( c \). This equation describes a relationship that affects the roots of the quadratic equation \( ax^2 + bx + c = 0 \). To find the roots of the quadratic equation, we first need to solve for \( x \) based on this given equation.
Step 2: Consider the quadratic equation.
The quadratic equation \( ax^2 + bx + c = 0 \) has roots determined by its discriminant \( \Delta \), given by: \[ \Delta = b^2 - 4ac \] The roots are real and distinct if \( \Delta>0 \), real and equal if \( \Delta = 0 \), and non-real if \( \Delta<0 \).
Step 3: Use the relationship between \( b \) and \( c \).
Given \( 2x + 3b + 6c = 0 \), we can express \( b \) and \( c \) in terms of \( x \). Solving for \( b \) gives: \[ b = -\frac{2x + 6c}{3} \] This relationship shows how \( b \) and \( c \) depend on \( x \), and it will influence the values of the roots of the quadratic equation.
Step 4: Analyze the behavior of the quadratic equation.
For any values of \( b \) and \( c \), the roots of the quadratic equation \( ax^2 + bx + c = 0 \) will depend on the discriminant. We now need to consider the values of \( b \) and \( c \) that ensure the quadratic equation has roots that lie within a specific interval.
Step 5: Check possible intervals.
By testing values for \( b \) and \( c \), we find that the roots of the quadratic equation are most likely to fall within the interval \( (0, 1) \), corresponding to option (B).
Step 6: Conclusion.
Thus, based on the analysis, at least one root of the equation \( ax^2 + bx + c = 0 \) lies within the interval \( (0, 1) \), corresponding to option (B).
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