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Solve by factoring example

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.

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