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The roots of the quadratic equation ax² + bx + c = 0 are real and equal. If the sum of the roots is equal to the sum of the coefficients a, b and c, what is the value of (a+b/2) whole square?
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The roots of the quadratic equation ax² + bx + c = 0 are real and equa...
Understanding the Problem
The given quadratic equation is ax² + bx + c = 0. For the roots to be real and equal, the discriminant must be zero:
- Discriminant: b² - 4ac = 0
This implies:
- b² = 4ac
Sum of the Roots
The sum of the roots (S) of the quadratic equation is given by:
- S = -b/a
According to the problem, this sum is equal to the sum of the coefficients:
- S = a + b + c
Setting these two expressions equal gives:
- -b/a = a + b + c
Finding the Coefficients Relationship
Rearranging the equation:
- -b = a(a + b + c)
Substituting b² = 4ac into this equation allows us to express one variable in terms of others.
Evaluating (a + b/2)²
From the equation b² = 4ac, we can express c in terms of a and b:
- c = b²/(4a)
Now, substituting this into the sum of coefficients:
- a + b + b²/(4a)
To find the value of (a + b/2)², expand it:
- (a + b/2)² = a² + ab + b²/4
With the conditions provided, and by solving the equation set previously, we can evaluate the expression.
Conclusion
After substituting and simplifying, the final answer emerges. The value of (a + b/2)² evaluates to:
- (a + b/2)² = (3/4)(b²/a)
This leads us to conclude that the relationships and conditions provided uniquely determine the values of a, b, and c based on their dependencies, leading to a calculable result.
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The roots of the quadratic equation ax² + bx + c = 0 are real and equal. If the sum of the roots is equal to the sum of the coefficients a, b and c, what is the value of (a+b/2) whole square?
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