To find the value of "b" in the equation x^2 + bx - 55 = 0, we can use the fact that the solutions of the quadratic equation are given as -11 and 5.
The solutions of a quadratic equation in the form ax^2 + bx + c = 0 can be found using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, we can substitute the given values of the solutions:
-11 = (-b ± √(b^2 - 4ac)) / (2a)
5 = (-b ± √(b^2 - 4ac)) / (2a)
Since both -11 and 5 are solutions, we can write two separate equations:
1) -11 = (-b + √(b^2 - 4ac)) / (2a)
2) 5 = (-b - √(b^2 - 4ac)) / (2a)
We can simplify these equations further:
1) -11(2a) = -b + √(b^2 - 4ac)
2) 5(2a) = -b - √(b^2 - 4ac)
Now let's solve for "b" by substituting the coefficients:
-22a = -b + √(b^2 + 220a)
10a = -b - √(b^2 + 220a)
To eliminate the square root, we can square both sides of both equations:
(-22a)^2 = (-b)^2 + 220a
(10a)^2 = (-b)^2 + 220a
484a^2 = b^2 + 220a
100a^2 = b^2 + 220a
Now let's solve these equations for "b":
484a^2 - b^2 = 220a
100a^2 - b^2 = 220a
After simplifying further:
b^2 - 484a^2 = -220a
b^2 - 100a^2 = -220a
Finally, we can isolate "b" by moving all terms to one side:
b^2 + 220a - 484a^2 = 0
b^2 + 220a - 100a^2 = 0
These equations represent two separate quadratic equations. We can solve them using factoring, completing the square, or the quadratic formula, depending on their factors. The solutions will give us the possible values of "b".
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