Math / Algebra

QuestionFactor the following expression completely: $3 w^{2}-19 w-14=$ $\square$ Question Help: Message instructor Submit Question
Question 10 Factor the following expression completely: $3 x^{2}-11 x-20=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to rewrite these expressions as a product of simpler expressions.
It's like reverse multiplication! Watch out! Don't forget to check if you can factor out a common factor from all terms at the very beginning.
It'll make your life much easier!
Also, double check your final factored form by multiplying it back out to make sure it matches the original expression.

STEP 2

1. Factor the first expression
2. Factor the second expression

STEP 3

We're starting with $3w^2 - 19w - 14$ .
Let's factor this bad boy!

STEP 4

First, we check if there's a common factor across all terms.
Nope, nothing we can pull out here.

STEP 5

Since the coefficient of $w^2$ is 3, and the constant term is -14, we look for two numbers that multiply to $3 \cdot -14 = -42$ and add up to the coefficient of our $w$ term, which is -19.

STEP 6

Let's think... $2$ and $-21$ work! $2 \cdot -21 = -42$ and $2 + (-21) = -19$ .
Perfect!

STEP 7

We rewrite the middle term using our magic numbers: $3w^2 - 19w - 14 = 3w^2 + 2w - 21w - 14$ .

STEP 8

Now, we factor by grouping: $3w^2 + 2w - 21w - 14 = w(3w + 2) - 7(3w + 2)$ .
Notice that $(3w + 2)$ is a common factor!

STEP 9

We factor out $(3w + 2)$ : $w(3w + 2) - 7(3w + 2) = (3w + 2)(w - 7)$ .
Boom!

STEP 10

Now, let's tackle $3x^2 - 11x - 20$ .

STEP 11

Again, no common factor to pull out initially.

STEP 12

The coefficient of $x^2$ is 3, and the constant term is -20, so we look for two numbers that multiply to $3 \cdot -20 = -60$ and add up to -11.

STEP 13

Hmm... $4$ and $-15$ fit the bill! $4 \cdot -15 = -60$ and $4 + (-15) = -11$ .
Awesome!

STEP 14

Rewrite the middle term: $3x^2 - 11x - 20 = 3x^2 + 4x - 15x - 20$ .

STEP 15

Factor by grouping: $3x^2 + 4x - 15x - 20 = x(3x + 4) - 5(3x + 4)$ .
We see $(3x + 4)$ is a common factor!

STEP 16

Factor out $(3x + 4)$ : $x(3x + 4) - 5(3x + 4) = (3x + 4)(x - 5)$ .
Nailed it!

STEP 17

The factored form of $3w^2 - 19w - 14$ is $(3w + 2)(w - 7)$ .
The factored form of $3x^2 - 11x - 20$ is $(3x + 4)(x - 5)$ .

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QuestionFactor the following expression completely: $3 w^{2}-19 w-14=$ $\square$ Question Help: Message instructor Submit Question
Question 10 Factor the following expression completely: $3 x^{2}-11 x-20=$ $\square$

Studdy Solution

STEP 1

STEP 2

1. Factor the first expression
2. Factor the second expression

STEP 3

We're starting with $3w^2 - 19w - 14$ .
Let's factor this bad boy!

STEP 4

First, we check if there's a common factor across all terms.
Nope, nothing we can pull out here.

STEP 5

Since the coefficient of $w^2$ is 3, and the constant term is -14, we look for two numbers that multiply to $3 \cdot -14 = -42$ and add up to the coefficient of our $w$ term, which is -19.

STEP 6

Let's think... $2$ and $-21$ work! $2 \cdot -21 = -42$ and $2 + (-21) = -19$ .
Perfect!

STEP 7

We rewrite the middle term using our magic numbers: $3w^2 - 19w - 14 = 3w^2 + 2w - 21w - 14$ .

STEP 8

Now, we factor by grouping: $3w^2 + 2w - 21w - 14 = w(3w + 2) - 7(3w + 2)$ .
Notice that $(3w + 2)$ is a common factor!

STEP 9

We factor out $(3w + 2)$ : $w(3w + 2) - 7(3w + 2) = (3w + 2)(w - 7)$ .
Boom!

STEP 10

Now, let's tackle $3x^2 - 11x - 20$ .

STEP 11

Again, no common factor to pull out initially.

STEP 12

The coefficient of $x^2$ is 3, and the constant term is -20, so we look for two numbers that multiply to $3 \cdot -20 = -60$ and add up to -11.

STEP 13

Hmm... $4$ and $-15$ fit the bill! $4 \cdot -15 = -60$ and $4 + (-15) = -11$ .
Awesome!

STEP 14

Rewrite the middle term: $3x^2 - 11x - 20 = 3x^2 + 4x - 15x - 20$ .

STEP 15

Factor by grouping: $3x^2 + 4x - 15x - 20 = x(3x + 4) - 5(3x + 4)$ .
We see $(3x + 4)$ is a common factor!

STEP 16

Factor out $(3x + 4)$ : $x(3x + 4) - 5(3x + 4) = (3x + 4)(x - 5)$ .
Nailed it!

STEP 17

The factored form of $3w^2 - 19w - 14$ is $(3w + 2)(w - 7)$ .
The factored form of $3x^2 - 11x - 20$ is $(3x + 4)(x - 5)$ .

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QuestionFactor the following expression completely: 3w2−19w−14=3 w^{2}-19 w-14=3w2−19w−14= □\square□ Question Help: Message instructor Submit QuestionQuestion 10 Factor the following expression completely: 3x2−11x−20=3 x^{2}-11 x-20=3x2−11x−20= □\square□

Studdy Solution

STEP 1

STEP 2

1. Factor the first expression2. Factor the second expression

STEP 3

We're starting with 3w2−19w−143w^2 - 19w - 143w2−19w−14.Let's **factor** this bad boy!

STEP 4

First, we check if there's a **common factor** across all terms.Nope, nothing we can pull out here.

STEP 5

Since the **coefficient** of w2w^2w2 is **3**, and the **constant term** is **-14**, we look for two numbers that multiply to 3⋅−14=−423 \cdot -14 = -423⋅−14=−42 and add up to the **coefficient** of our www term, which is **-19**.

STEP 6

Let's think... 222 and −21-21−21 work! 2⋅−21=−422 \cdot -21 = -422⋅−21=−42 and 2+(−21)=−192 + (-21) = -192+(−21)=−19.Perfect!

STEP 7

We **rewrite** the middle term using our magic numbers: 3w2−19w−14=3w2+2w−21w−143w^2 - 19w - 14 = 3w^2 + 2w - 21w - 143w2−19w−14=3w2+2w−21w−14.

STEP 8

Now, we **factor by grouping**: 3w2+2w−21w−14=w(3w+2)−7(3w+2)3w^2 + 2w - 21w - 14 = w(3w + 2) - 7(3w + 2)3w2+2w−21w−14=w(3w+2)−7(3w+2).Notice that (3w+2)(3w + 2)(3w+2) is a **common factor**!

STEP 9

We **factor out** (3w+2)(3w + 2)(3w+2): w(3w+2)−7(3w+2)=(3w+2)(w−7)w(3w + 2) - 7(3w + 2) = (3w + 2)(w - 7)w(3w+2)−7(3w+2)=(3w+2)(w−7).Boom!

STEP 10

Now, let's tackle 3x2−11x−203x^2 - 11x - 203x2−11x−20.

STEP 11

Again, no **common factor** to pull out initially.

STEP 12

The **coefficient** of x2x^2x2 is **3**, and the **constant term** is **-20**, so we look for two numbers that multiply to 3⋅−20=−603 \cdot -20 = -603⋅−20=−60 and add up to **-11**.

STEP 13

Hmm... 444 and −15-15−15 fit the bill! 4⋅−15=−604 \cdot -15 = -604⋅−15=−60 and 4+(−15)=−114 + (-15) = -114+(−15)=−11.Awesome!

STEP 14

**Rewrite** the middle term: 3x2−11x−20=3x2+4x−15x−203x^2 - 11x - 20 = 3x^2 + 4x - 15x - 203x2−11x−20=3x2+4x−15x−20.

STEP 15

**Factor by grouping**: 3x2+4x−15x−20=x(3x+4)−5(3x+4)3x^2 + 4x - 15x - 20 = x(3x + 4) - 5(3x + 4)3x2+4x−15x−20=x(3x+4)−5(3x+4).We see (3x+4)(3x + 4)(3x+4) is a **common factor**!

STEP 16

**Factor out** (3x+4)(3x + 4)(3x+4): x(3x+4)−5(3x+4)=(3x+4)(x−5)x(3x + 4) - 5(3x + 4) = (3x + 4)(x - 5)x(3x+4)−5(3x+4)=(3x+4)(x−5).Nailed it!

STEP 17

The factored form of 3w2−19w−143w^2 - 19w - 143w2−19w−14 is (3w+2)(w−7)(3w + 2)(w - 7)(3w+2)(w−7).The factored form of 3x2−11x−203x^2 - 11x - 203x2−11x−20 is (3x+4)(x−5)(3x + 4)(x - 5)(3x+4)(x−5).

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QuestionFactor the following expression completely: $3 w^{2}-19 w-14=$ $\square$ Question Help: Message instructor Submit Question
Question 10 Factor the following expression completely: $3 x^{2}-11 x-20=$ $\square$

1. Factor the first expression
2. Factor the second expression

We're starting with $3w^2 - 19w - 14$ .
Let's factor this bad boy!

First, we check if there's a common factor across all terms.
Nope, nothing we can pull out here.

Since the coefficient of $w^2$ is 3, and the constant term is -14, we look for two numbers that multiply to $3 \cdot -14 = -42$ and add up to the coefficient of our $w$ term, which is -19.

Let's think... $2$ and $-21$ work! $2 \cdot -21 = -42$ and $2 + (-21) = -19$ .
Perfect!

We rewrite the middle term using our magic numbers: $3w^2 - 19w - 14 = 3w^2 + 2w - 21w - 14$ .

Now, we factor by grouping: $3w^2 + 2w - 21w - 14 = w(3w + 2) - 7(3w + 2)$ .
Notice that $(3w + 2)$ is a common factor!

We factor out $(3w + 2)$ : $w(3w + 2) - 7(3w + 2) = (3w + 2)(w - 7)$ .
Boom!

Now, let's tackle $3x^2 - 11x - 20$ .

Again, no common factor to pull out initially.

The coefficient of $x^2$ is 3, and the constant term is -20, so we look for two numbers that multiply to $3 \cdot -20 = -60$ and add up to -11.

Hmm... $4$ and $-15$ fit the bill! $4 \cdot -15 = -60$ and $4 + (-15) = -11$ .
Awesome!

Rewrite the middle term: $3x^2 - 11x - 20 = 3x^2 + 4x - 15x - 20$ .

Factor by grouping: $3x^2 + 4x - 15x - 20 = x(3x + 4) - 5(3x + 4)$ .
We see $(3x + 4)$ is a common factor!

Factor out $(3x + 4)$ : $x(3x + 4) - 5(3x + 4) = (3x + 4)(x - 5)$ .
Nailed it!

The factored form of $3w^2 - 19w - 14$ is $(3w + 2)(w - 7)$ .
The factored form of $3x^2 - 11x - 20$ is $(3x + 4)(x - 5)$ .