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The polynomial p(x) = x^2+ax+b has distinct roots 2a and 2b. Find a+b.

 Jun 24, 2023
 #1
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-1

Since 2a and 2b are the roots of p(x), we know that

p(x) = (x - 2a)(x - 2b) = x^2 - 4ax + 4ab

Equating coefficients, we get

a = -4a b = 4ab

Solving for a and b, we find

a = -1 b = -2

Therefore, a+b = -3.

 Jun 24, 2023
 #2
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-1

Thank you... but the answer is actually -1/4.. still appreciate the effort!

 Jun 24, 2023
 #3
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To find the values of a and b, we can use the relationship between the roots and coefficients of a quadratic equation.

In a quadratic equation of the form p(x) = x^2 + ax + b, the sum of the roots is equal to the negation of the coefficient of x (a), and the product of the roots is equal to the constant term (b).

Given that the distinct roots are 2a and 2b, we have the following relationships:

Sum of the roots: 2a + 2b = -a (1) Product of the roots: (2a)(2b) = b (2)

From equation (1), we can simplify it by moving all terms to one side:

2a + 2b + a = 0 3a + 2b = 0

Now, let's solve equations (1) and (2) simultaneously:

From equation (2), we have (2a)(2b) = b, which can be rewritten as 4ab = b. We can divide both sides by b (assuming b is nonzero) to obtain 4a = 1. Dividing both sides by 4, we get a = 1/4.

Substituting the value of a = 1/4 into equation (1): myhdfs login

3(1/4) + 2b = 0 3/4 + 2b = 0 2b = -3/4 b = -3/8

Therefore, the values of a and b are a = 1/4 and b = -3/8, respectively. To find a + b, we can substitute these values:

a + b = 1/4 + (-3/8) = 2/8 - 3/8 = -1/8.

Hence, the value of a + b is -1/8.

 Jun 24, 2023

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