View the Factoring 101 post to get started with the basics of factoring first!
Q: Solve by factoring: x² – 7x = 78
A: When solving by factoring, the first thing to do is set the equation equal to zero:
x² – 7x = 78 [subtract 78 from both sides]
x² – 7x – 78 = 0
Now, to factor… We want two set up the equation like so:
( )( ) = 0
We know the first term in each factor (or set of parentheses) is “x” because they must multiply to get x².
(1) ( x )( x ) = 0
Now, I like to figure out the sings. The remaining numbers must multiply to give -78 and combine (by addition or subtraction) to get -7. Just paying attention to the signs. In order to multiple to get a negative number (-78), we would need a positive and a negative sign, like so:
(2) ( x – )( x + ) = 0
OK. Almost there. Now we think: What two numbers multiply to give -78 and combine to give -7? Some factors of -78 are -2 & 39, -6 & 13, 6 & -13, 1 & -78, and many more options! Factoring is an art. You must be good at guessing and visualizing to save time! Notice that -13 and 6 combine to give -7? Great news! So, we have:
(3) ( x – 13)( x + 6) = 0
Indeed, if you used the distributive property, or as we like to say: FOIL: you would find that ( x – 13)( x + 6) would multiply out to give x² – 7x – 78 (this is just a way to check our work).
OK… If ( x – 13)( x + 6) = 0, then either
( x – 13) = 0 or ( x + 6) = 0.
So, x = 13 or x = -6.