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If b and 7 are solutions to the equation
x² - 4x = a. find the value of a + b.
x² - 4x = a
is a quadratic equation, and has at most 2 solutions.
It is equivalent to
x² - 4x - a = 0
Since its two solutions are b and 7, then the left side
must factor as
(x - b)(x - 7) = 0
x² - 7x - bx + 7b = 0
Factor x out of the two middle terms:
x² + x(-7 - b) + 7b = 0
x² + (-7 - b)x + 7b = 0
Comparing that to
x² - 4x - a = 0
We equate the coefficients of x
-7 - b = -4
Solving that:
-b = -4+7
-b = 3
b = -3
We also equate the constant terms:
-a = 7b
Substituting -3 for b in that
-a = 7(-3)
-a = -21
a = 21
So a + b = 21 + (-3) = 18
Edwin
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