Studdy AI Logo
Math  /  Algebra

Questiona2+12a+35a2+2a−48⋅a2+3a−54a2+14a+49=\frac{a^{2}+12 a+35}{a^{2}+2 a-48} \cdot \frac{a^{2}+3 a-54}{a^{2}+14 a+49}= (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We're asked to multiply two fractions with variables and simplify the result! Watch out! Remember to factor everything completely before simplifying, and don't forget about those sneaky signs!

STEP 2

1. Factor the Numerator of the First Fraction
2. Factor the Denominator of the First Fraction
3. Factor the Numerator of the Second Fraction
4. Factor the Denominator of the Second Fraction
5. Multiply the Fractions
6. Simplify the Expression

STEP 3

Let's **factor** a2+12a+35a^2 + 12a + 35.
We're looking for two numbers that **add up** to 1212 and **multiply** to 3535.
Those magic numbers are 55 and 77!
So, a2+12a+35=(a+5)(a+7)a^2 + 12a + 35 = (a+5)(a+7).
Awesome!

STEP 4

Now, let's **factor** a2+2a−48a^2 + 2a - 48.
We need two numbers that **add** to 22 and **multiply** to −48-48.
After a little thought, we find that 88 and −6-6 fit the bill.
So, a2+2a−48=(a+8)(a−6)a^2 + 2a - 48 = (a+8)(a-6).
Fantastic!

STEP 5

Time to **factor** a2+3a−54a^2 + 3a - 54.
We're searching for two numbers that **add** to 33 and **multiply** to −54-54.
A-ha! 99 and −6-6 work perfectly.
So, a2+3a−54=(a+9)(a−6)a^2 + 3a - 54 = (a+9)(a-6).
Keep it going!

STEP 6

Let's **factor** a2+14a+49a^2 + 14a + 49.
We need two numbers that **add** to 1414 and **multiply** to 4949.
That's 77 and 77!
So, a2+14a+49=(a+7)(a+7)a^2 + 14a + 49 = (a+7)(a+7), which can also be written as (a+7)2(a+7)^2.
Almost there!

STEP 7

Now, let's put it all together and **multiply** the fractions: (a+5)(a+7)(a+8)(a−6)⋅(a+9)(a−6)(a+7)(a+7) \frac{(a+5)(a+7)}{(a+8)(a-6)} \cdot \frac{(a+9)(a-6)}{(a+7)(a+7) }

STEP 8

Time to **simplify**!
We can divide to one the (a+7)(a+7) and (a−6)(a-6) terms that appear in both the numerator and the denominator.
Remember, we're not "canceling" them, we're dividing them to one! (a+5)(a+7)(a+8)(a−6)⋅(a+9)(a−6)(a+7)(a+7)=(a+5)(a+9)(a+8)(a+7) \frac{(a+5)(a+7)}{(a+8)(a-6)} \cdot \frac{(a+9)(a-6)}{(a+7)(a+7)} = \frac{(a+5)(a+9)}{(a+8)(a+7)}

STEP 9

Our **simplified expression** is: (a+5)(a+9)(a+8)(a+7)\frac{(a+5)(a+9)}{(a+8)(a+7)}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord