QuestionFactor the trinomial: $2x^{2} + 21x + 49$ .

Studdy Solution

STEP 1

Assumptions1. We are given a trinomial of the form $ax^ + bx + c$ where $a =$ , $b =21$ , and $c =49$ . . We are asked to factor this trinomial.

STEP 2

A trinomial of the form $ax^2 + bx + c$ can be factored into the form $(dx + e)(fx + g)$ , where $df = a$ , $eg = c$ , and $dg + ef = b$ .

STEP 3

We need to find values for $d$ , $e$ , $f$ , and $g$ that satisfy these conditions.

STEP 4

Start by finding two numbers that multiply to $ac =249 =98$ and add to $b =21$ .

STEP 5

The numbers that satisfy these conditions are $7$ and $14$ because $7*14 =98$ and $7 +14 =21$ .

STEP 6

Now we can rewrite the middle term of the trinomial using these two numbers.
$2x^2 +x +14x +49$

STEP 7

Factor by grouping. First group the first two terms together and the last two terms together.
$(2x^2 +7x) + (14x +49)$

STEP 8

Factor out the greatest common factor from each group.
$x(2x +7) +7(2x +7)$

STEP 9

Notice that $(2x +7)$ is a common factor. Factor this out.
$(2x +7)(x +7)$ So, the factored form of the trinomial $2x^2 +21x +49$ is $(2x +7)(x +7)$ .

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