Question

If the roots of $a x^{2}+b x+c=0$ are $2, \frac{3}{2}$ then $(a+b+c)^{2}$

The answer is $\frac{1}{8}$.

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Answer

To solve this problem, we start with the given quadratic equation:

[ a x^2 + b x + c = 0 ]

We know the roots of this equation are (2) and (\frac{3}{2}). This allows us to express the equation in its factored form:

[ (x - 2)\left(x - \frac{3}{2}\right) = 0 ]

Next, we expand and simplify the factored form:

[ (x - 2)\left(x - \frac{3}{2}\right) = x^2 - \frac{3}{2} x - 2x + 3 = x^2 - \frac{7}{2} x + 3 ]

From the comparison with the general quadratic equation (a x^2 + b x + c = 0), we can identify:

[ a = 1, \quad b = -\frac{7}{2}, \quad \text{and} \quad c = 3 ]

We are asked to find ((a + b + c)^2). First, we sum the values of (a), (b), and (c):

[ a + b + c = 1 + \left(-\frac{7}{2}\right) + 3 ]

To simplify this, we find a common denominator:

[ a + b + c = 1 - \frac{7}{2} + 3 = \frac{2}{2} - \frac{7}{2} + \frac{6}{2} = \frac{2 - 7 + 6}{2} = \frac{1}{2} ]

Finally, we square this sum:

[ (a + b + c)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} ]

Thus, the correct answer is:

[ \boxed{\frac{1}{4}} ]


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