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# HELP

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The roots of the quadratic equation \(x^2 + bx + c = 0 \) are \(5 + 3i\) and \(5 - 3i\). What is \(b+c\)?

Apr 25, 2024

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We can find the values of b and c using the relationship between the quadratic formula and the roots of the equation.

The quadratic formula relates the coefficients (a, b, and c) of the quadratic equation to its roots (r1 and r2):

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, we know the roots (5 + 3i and 5 - 3i) and want to find b + c.

Properties of Complex Roots:

Complex numbers as roots of quadratic equations always come in conjugate pairs. This means that if one root is a + bi (where i is the imaginary unit), the other root will be a - bi.

In this case, our roots are 5 + 3i and 5 - 3i, which confirms they are complex conjugates.

Sum of Roots and Coefficients:

There's a useful relationship between the roots (r1 and r2) of a quadratic equation and its coefficients (a, b, and c):

Sum of roots (r1 + r2) = -b / a

Product of roots (r1 * r2) = c / a

Since we're looking for b + c, and the quadratic has a leading coefficient of 1 (a = 1), we can use these relationships directly.

Finding b + c:

Sum of Roots:

The sum of the roots (5 + 3i) and (5 - 3i) is:

(5 + 5) + (3i - 3i) = 10

Since the sum of roots is also -b / a (and a = 1), we have:

-b = 10

Therefore, b = -10

Product of Roots:

The product of the roots (5 + 3i) and (5 - 3i) is:

(5 + 3i) * (5 - 3i) = 25 + 25 = 50

Since the product of roots is also c / a (and a = 1), we have:

c = 50

b + c:

Finally, add b and c to find the desired value:

b + c = -10 + 50 = 40

Therefore, b + c = 40.

May 9, 2024