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Factoring Trinomials of Type ax2 +bx +c, a la Shmoop

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Farmer Jen Etticks produces donut-shaped watermelons. [Farmer carrying a watermelon]

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Like Godzilla, they’re huge in Japan.

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There’s just one problem: tossing the spoiled watermelons into the ocean has had some… [Farmer throws watermelon into the ocean]

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undesirable effects.

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The guy in accounting says that the amount of spoilage, or “s”…

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…for producing w number of watermelons per day can be approximated by the formula: [accounting guy discussing formula]

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s equals 6w squared plus 13w plus 2.

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What daily production level will give Jen zero spoilage?

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This looks like a trinomial equation of the type ax squared plus bx plus c; [man points to trinomial equation]

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To find the answer, we’re going to need to factor out the a and the c in the equation.

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We’ll do this by using a table. [Dinosaur jumps on a table]

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Our table has four columns.

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The first involves our “a-term”, 6w squared.

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There are two possible ways to factor 6w squared: 3w and 2w, and 6w and w.

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The second column factors our “c-term”, 2.

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The only possible factors are 2 and 1.

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The third and fourth columns use the previous two columns to decipher possible “b-terms”.

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Combinations of the a-term and c-term factors are combined in the “possible binomial factors” [arrow pointing to binomial factors column]

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column.

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If we take a look at the first row of the first two columns, our possible binomial factors

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come out to:

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(3w + 2) times (2w + 1), and (3w+1) times (2w+2).

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For the second row, we get (6w +2) times (w+1), and (6w+1) times (2+2).

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Then, the sum of the inner and outer products are used to calculate what the b-term for

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that product would be.

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So for (3w +2) times (2w + 1), we take the outer product first, which is 3w times 1, [table of factoring trinomials]

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and add it to the inner products: 2w times 2.

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3w times 1 is 3w, and 2w times 2 is 4w.

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So the sum is 7w.

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Not the right b-term...so we keep trying. [man points to b-term]

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After doing all the calculations, we can see that the factors (6w + 2) and (w + 2) give

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us our correct b-term, 13w, hiding out like a bunny at a fox convention. [fox hiding among bunnies]

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Our work can be checked using FOIL, multiplying out the first terms, outer terms, inner terms,

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and last terms,.

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We add them together…and check to see that they equal our original trinomial. [man looking at the completed formula]

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We passed the checkpoint, which means when spoilage equals zero, one of our factors must

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also equal zero.

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When we set each factor to zero and solve for w…

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…we see there is zero spoilage when our daily watermelon production equals negative

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one sixth and negative 2. [doughnut watermelons appear]

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Uh-oh.

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Since it’s impossible to have negative production…

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…looks like Farmer Jen has no choice but to introduce spoiled watermelons into the [farmer Jen throwing a watermelon into the ocean]

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food supply.

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Good luck, everyone.

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