Solved on Jan 23, 2024

Simplify the expression $9 w^{2} + 9 w + 2$ .

STEP 1

Assumptions
1. We are given a quadratic expression $9w^2 + 9w + 2$ .
2. The task is to solve the quadratic equation by finding the roots of the expression, if that is the intention of the problem. If the task is to factor the expression, we will proceed with factoring.

STEP 2

First, we need to determine whether the quadratic expression can be factored into the form $(aw + b)(cw + d)$ , where $a$ , $b$ , $c$ , and $d$ are constants.

STEP 3

To factor the quadratic expression, we look for two numbers that multiply to $9w^2 \times 2$ (the product of the coefficient of $w^2$ and the constant term) and add up to $9w$ (the coefficient of the $w$ term).

STEP 4

The product of the coefficient of $w^2$ and the constant term is $9 \times 2 = 18$ .

STEP 5

We need to find two numbers that multiply to $18$ and add up to $9$ .

STEP 6

The numbers that satisfy these conditions are $6$ and $3$ because $6 \times 3 = 18$ and $6 + 3 = 9$ .

STEP 7

We rewrite the middle term $9w$ using the numbers $6$ and $3$ .
$9w^2 + 9w + 2 = 9w^2 + 6w + 3w + 2$

STEP 8

Now we group the terms to factor by grouping.
$9w^2 + 6w + 3w + 2 = (9w^2 + 6w) + (3w + 2)$

STEP 9

Factor out the common factors from each group.
$9w^2 + 6w = 3w(3w + 2)$ $3w + 2 = 1(3w + 2)$

STEP 10

Now we have two terms with a common factor of $(3w + 2)$ .
$3w(3w + 2) + 1(3w + 2) = (3w + 2)(3w + 1)$

STEP 11

The quadratic expression $9w^2 + 9w + 2$ is now factored into $(3w + 2)(3w + 1)$ .
The factored form of the quadratic expression is $(3w + 2)(3w + 1)$ .

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Simplify the expression $9 w^{2} + 9 w + 2$ .

STEP 1

Assumptions
1. We are given a quadratic expression $9w^2 + 9w + 2$ .
2. The task is to solve the quadratic equation by finding the roots of the expression, if that is the intention of the problem. If the task is to factor the expression, we will proceed with factoring.

STEP 2

First, we need to determine whether the quadratic expression can be factored into the form $(aw + b)(cw + d)$ , where $a$ , $b$ , $c$ , and $d$ are constants.

STEP 3

To factor the quadratic expression, we look for two numbers that multiply to $9w^2 \times 2$ (the product of the coefficient of $w^2$ and the constant term) and add up to $9w$ (the coefficient of the $w$ term).

STEP 4

The product of the coefficient of $w^2$ and the constant term is $9 \times 2 = 18$ .

STEP 5

We need to find two numbers that multiply to $18$ and add up to $9$ .

STEP 6

The numbers that satisfy these conditions are $6$ and $3$ because $6 \times 3 = 18$ and $6 + 3 = 9$ .

STEP 7

We rewrite the middle term $9w$ using the numbers $6$ and $3$ .
$9w^2 + 9w + 2 = 9w^2 + 6w + 3w + 2$

STEP 8

Now we group the terms to factor by grouping.
$9w^2 + 6w + 3w + 2 = (9w^2 + 6w) + (3w + 2)$

STEP 9

Factor out the common factors from each group.
$9w^2 + 6w = 3w(3w + 2)$ $3w + 2 = 1(3w + 2)$

STEP 10

Now we have two terms with a common factor of $(3w + 2)$ .
$3w(3w + 2) + 1(3w + 2) = (3w + 2)(3w + 1)$

STEP 11

The quadratic expression $9w^2 + 9w + 2$ is now factored into $(3w + 2)(3w + 1)$ .
The factored form of the quadratic expression is $(3w + 2)(3w + 1)$ .

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Simplify the expression 9w2+9w+29 w^{2} + 9 w + 29w2+9w+2.

STEP 1

Assumptions1. We are given a quadratic expression 9w2+9w+29w^2 + 9w + 29w2+9w+2.2. The task is to solve the quadratic equation by finding the roots of the expression, if that is the intention of the problem. If the task is to factor the expression, we will proceed with factoring.

STEP 2

First, we need to determine whether the quadratic expression can be factored into the form (aw+b)(cw+d)(aw + b)(cw + d)(aw+b)(cw+d), where aaa, bbb, ccc, and ddd are constants.

STEP 3

To factor the quadratic expression, we look for two numbers that multiply to 9w2×29w^2 \times 29w2×2 (the product of the coefficient of w2w^2w2 and the constant term) and add up to 9w9w9w (the coefficient of the www term).

STEP 4

The product of the coefficient of w2w^2w2 and the constant term is 9×2=189 \times 2 = 189×2=18.

STEP 5

We need to find two numbers that multiply to 181818 and add up to 999.

STEP 6

The numbers that satisfy these conditions are 666 and 333 because 6×3=186 \times 3 = 186×3=18 and 6+3=96 + 3 = 96+3=9.

STEP 7

We rewrite the middle term 9w9w9w using the numbers 666 and 333.9w2+9w+2=9w2+6w+3w+29w^2 + 9w + 2 = 9w^2 + 6w + 3w + 29w2+9w+2=9w2+6w+3w+2

STEP 8

Now we group the terms to factor by grouping.9w2+6w+3w+2=(9w2+6w)+(3w+2)9w^2 + 6w + 3w + 2 = (9w^2 + 6w) + (3w + 2)9w2+6w+3w+2=(9w2+6w)+(3w+2)

STEP 9

Factor out the common factors from each group.9w2+6w=3w(3w+2)9w^2 + 6w = 3w(3w + 2)9w2+6w=3w(3w+2) 3w+2=1(3w+2)3w + 2 = 1(3w + 2)3w+2=1(3w+2)

STEP 10

Now we have two terms with a common factor of (3w+2)(3w + 2)(3w+2).3w(3w+2)+1(3w+2)=(3w+2)(3w+1)3w(3w + 2) + 1(3w + 2) = (3w + 2)(3w + 1)3w(3w+2)+1(3w+2)=(3w+2)(3w+1)

STEP 11

The quadratic expression 9w2+9w+29w^2 + 9w + 29w2+9w+2 is now factored into (3w+2)(3w+1)(3w + 2)(3w + 1)(3w+2)(3w+1).The factored form of the quadratic expression is (3w+2)(3w+1)(3w + 2)(3w + 1)(3w+2)(3w+1).

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Simplify the expression $9 w^{2} + 9 w + 2$ .

Assumptions
1. We are given a quadratic expression $9w^2 + 9w + 2$ .
2. The task is to solve the quadratic equation by finding the roots of the expression, if that is the intention of the problem. If the task is to factor the expression, we will proceed with factoring.

First, we need to determine whether the quadratic expression can be factored into the form $(aw + b)(cw + d)$ , where $a$ , $b$ , $c$ , and $d$ are constants.

To factor the quadratic expression, we look for two numbers that multiply to $9w^2 \times 2$ (the product of the coefficient of $w^2$ and the constant term) and add up to $9w$ (the coefficient of the $w$ term).

The product of the coefficient of $w^2$ and the constant term is $9 \times 2 = 18$ .

We need to find two numbers that multiply to $18$ and add up to $9$ .

The numbers that satisfy these conditions are $6$ and $3$ because $6 \times 3 = 18$ and $6 + 3 = 9$ .

We rewrite the middle term $9w$ using the numbers $6$ and $3$ .
$9w^2 + 9w + 2 = 9w^2 + 6w + 3w + 2$

Now we group the terms to factor by grouping.
$9w^2 + 6w + 3w + 2 = (9w^2 + 6w) + (3w + 2)$

Factor out the common factors from each group.
$9w^2 + 6w = 3w(3w + 2)$ $3w + 2 = 1(3w + 2)$

Now we have two terms with a common factor of $(3w + 2)$ .
$3w(3w + 2) + 1(3w + 2) = (3w + 2)(3w + 1)$

The quadratic expression $9w^2 + 9w + 2$ is now factored into $(3w + 2)(3w + 1)$ .
The factored form of the quadratic expression is $(3w + 2)(3w + 1)$ .