The sum and the zeroes of quadratic polynomial px^2 + 2x + 3q is equal...
Explanation:
Let the zeroes of the quadratic polynomial px^2 + 2x + 3q be α and β.
Then, sum of the zeroes = α + β
Product of the zeroes = αβ
Given, sum of the zeroes + sum of the coefficients = product of the zeroes.
α + β + 2 + 3q = αβ
Rearranging the terms, we get:
αβ - α - β - 2 - 3q = 0
Now, we can rewrite the left-hand side of the equation as a product of two linear factors:
(α - 1)(β - 1) - 3(q + 1) = 0
Since the product of two linear factors is zero if and only if at least one of the factors is zero, we have two cases to consider:
Case 1: α - 1 = 0
This implies that α = 1, and substituting this into the equation α + β + 2 + 3q = αβ gives us:
1 + β + 2 + 3q = β
Simplifying, we get:
3q - 1 = 0
Therefore, q = 1/3.
Case 2: β - 1 = 0
This implies that β = 1, and substituting this into the equation α + β + 2 + 3q = αβ gives us:
α + 1 + 2 + 3q = α
Simplifying, we get:
3q + 3 = 0
Therefore, q = -1.
Conclusion:
Thus, we have found that the value of q can either be 1/3 or -1. However, we cannot determine the value of p from the given information, as there are infinitely many quadratic polynomials that satisfy the condition that their sum and zeroes are equal to their product of zeroes.
The sum and the zeroes of quadratic polynomial px^2 + 2x + 3q is equal...
P(0)=2 +2(0) + 3q =2+ 0 3q =2+ 3q =3q = ‐2 q = ‐2/3 So, the value of "p" is ‐2/3
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