x² - 15 = 2x
Get 0 on the right by adding -2x to both sides.
x² - 15 - 2x = 0
Arrange the left side in descending order of
eponents of x:
We need to factor the left side:
Give the x² the coefficient of 1
1x² - 2x - 15 = 0
Multiply the red 1 by the blue 15, getting 15.
Now think of a pair of positive integers whose product
is the blue 15 and whose difference is the green 2.
I said difference because the last sign (before the
15) is minus. Had it been plus, I would have said "sum".
We think of the integers 3 and 5 because their product is
the blue 15 and their difference is the green 2.
Now we use the 3 and 5 to rewrite the green -2 as 3 - 5.
So we rewrite 2x as -3x + 5x. That is,
x² - 2x - 15 = 0
becomes
x² + 3x - 5x - 15 = 0
Now we factor x out of the first two terms
x(x + 3) - 5x - 15 = 0
and factor -5 out of the last two terms on the
left:
x(x + 3) - 5(x + 3) = 0
Be careful to notice that when we factor a
NEGATIVE number, -5, out of another NEGATIVE
number -15, we get a POSITIVE 3.
Now we have a common factor, which we can
factor out, namely the (x + 3)'s which I
color red:
x(x + 3) - 5(x + 3) = 0
Factor out the red parentheses:
(x + 3)(x - 5) = 0
Setting the first factor (x + 3) = 0 gives x = -3
Setting the second factor (x - 5) = 0 gives x = 5
So the solutions are -3 and 5.
Edwin
