+180 The equation x2+ ax + 2023 = 0 has roots x1 = b + 1 and x2 = 8b2 + 1 for some real number b. Find a + b.
So far I have been trying Vieta's and getting 8b3 + 8b2 + b +1 = 2023, -a = 8b2 + b + 2. And then getting 2023 + a = 8b^3, but then I don't know what to do from here, is my approach totally wrong? Also, can I have the answer with a solution, so I understand how you got the answer?
The sum of the roots of a quadratic equation is given by -b/a, and the product of the roots is given by c/a. In this case, the sum of the roots is x1 + x2 = b + 1 + 8b^2 + 1 = 8b^2 + b + 2. The product of the roots is x1*x2 = (b + 1)(8b^2 + 1) = 8b^3 + b + 1.
We are given that the sum of the roots is -a/1 and the product of the roots is 2023/1. Setting these equal to the expressions we just derived, we get the following equations:
-a/1 = 8b^2 + b + 2 2023/1 = 8b^3 + b + 1
Multiplying the first equation by 1 and the second equation by -1, we get the following equations:
-a = 8b^2 + b + 2 -2023 = -8b^3 - b - 1
Adding these equations, we get the following equation:
-a - 2023 = -8b^3 - 8b^2 - 3
Combining like terms, we get the following equation:
-a - 2023 = -8b^3 - 8b^2 - 3
Multiplying both sides of the equation by -1, we get the following equation:
a + 2023 = 8b^3 + 8b^2 + 3
Now, we need to find the value of b. We can do this by using the fact that the sum of the roots of a quadratic equation is -b/a. In this case, the sum of the roots is x1 + x2 = b + 1 + 8b^2 + 1 = 8b^2 + b + 2. We are given that this value is equal to 2023. Setting these equal to each other, we get the following equation:
2023 = 8b^2 + b + 2
Subtracting 2 from both sides of the equation, we get the following equation:
2021 = 8b^2 + b
Combining like terms, we get the following equation:
2021 = 8b^2 + b
Factoring out a 1 from the right-hand side of the equation, we get the following equation:
2021 = 1(8b^2 + b)
Dividing both sides of the equation by 1, we get the following equation:
2021 = 8b^2 + b
Now, we can substitute this value of b into the equation a + 2023 = 8b^3 + 8b^2 + 3. Doing so, we get the following equation:
a + 2023 = 8(2021) + 8(1) + 3
Simplifying the right-hand side of the equation, we get the following equation:
a + 2023 = 16168 + 8 + 3
Combining like terms, we get the following equation:
a + 2023 = 16179
Subtracting 2023 from both sides of the equation, we get the following equation:
a = 16179 - 2023
Simplifying the right-hand side of the equation, we get the following equation:
a = 14156
Therefore, the value of a + b is 14156 + 2023 = 16179.
