Considering the sum and the product of the roots of the quadratic equation, it is found that the numeric value of the expression is given as follows:
![\alpha + \beta = -2.5](https://tex.z-dn.net/?f=%5Calpha%20%2B%20%5Cbeta%20%3D%20-2.5)
<h3>What are the sum and the product of the roots of a quadratic equation?</h3>
A quadratic equation is defined as follows:
![y = ax^2 + bx + c, a \neq 0](https://tex.z-dn.net/?f=y%20%3D%20ax%5E2%20%2B%20bx%20%2B%20c%2C%20a%20%5Cneq%200)
The roots of the equation are given as follows:
![\alpha, \beta](https://tex.z-dn.net/?f=%5Calpha%2C%20%5Cbeta)
The sum of the roots is given as follows:
![\alpha + \beta = -\frac{b}{a}](https://tex.z-dn.net/?f=%5Calpha%20%2B%20%5Cbeta%20%3D%20-%5Cfrac%7Bb%7D%7Ba%7D)
The product of the roots is given as follows:
![\alpha\beta = \frac{c}{a}](https://tex.z-dn.net/?f=%5Calpha%5Cbeta%20%3D%20%5Cfrac%7Bc%7D%7Ba%7D)
In the context of this problem, the product is of 4, as
hence:
c/a = 4
c = 4a.
The coefficients are in an arithmetic progression, hence:
- b = a + d. (d is the common difference of the sequence).
We have that c = 4a, hence:
4a = a + 2d
2d = 3a
d = 1.5a.
Hence coefficient b is calculated as follows:
b = a + d = a + 1.5a = 2.5a.
Then the sum of the roots is given as follows:
![\alpha + \beta = -\frac{b}{a} = -\frac{2.5a}{a} = -2.5](https://tex.z-dn.net/?f=%5Calpha%20%2B%20%5Cbeta%20%3D%20-%5Cfrac%7Bb%7D%7Ba%7D%20%3D%20-%5Cfrac%7B2.5a%7D%7Ba%7D%20%3D%20-2.5)
More can be learned about quadratic equations at brainly.com/question/24737967
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