Solved on Jan 12, 2024

Simplify $(2 x+3)^{2}$ and subtract it from $2 x^{2}-4$ to get the result in standard polynomial form.

STEP 1

Assumptions
1. We are given the expression $2 x^{2}-4$ .
2. We are also given the expression $(2 x+3)^{2}$ .
3. We need to subtract the second expression from the first one.
4. The result should be written as a simplified polynomial in standard form.
5. Standard form for a polynomial is written with terms in descending order of their degrees.
6. There is a penalty of 0.25 off for incorrect answers, but this does not affect the mathematical solution.

STEP 2

First, we need to expand the expression $(2 x+3)^{2}$ .
$(2 x+3)^{2} = (2 x+3)(2 x+3)$

STEP 3

Now, use the distributive property (FOIL method) to expand the product.
$(2 x+3)(2 x+3) = 2 x \cdot 2 x + 2 x \cdot 3 + 3 \cdot 2 x + 3 \cdot 3$

STEP 4

Perform the multiplication for each term.
$2 x \cdot 2 x = 4 x^{2}$ $2 x \cdot 3 = 6 x$ $3 \cdot 2 x = 6 x$ $3 \cdot 3 = 9$

STEP 5

Combine the like terms from the expansion.
$4 x^{2} + 6 x + 6 x + 9 = 4 x^{2} + 12 x + 9$

STEP 6

Now, subtract the expanded expression from the first expression given.
$(2 x^{2}-4) - (4 x^{2} + 12 x + 9)$

STEP 7

Distribute the subtraction across the terms in the second expression.
$2 x^{2}-4 - 4 x^{2} - 12 x - 9$

STEP 8

Combine like terms by subtracting the coefficients of the terms with the same degree.
For the $x^{2}$ terms: $2 x^{2} - 4 x^{2} = -2 x^{2}$
For the $x$ terms: Since there is no $x$ term in the first expression, we only have the $-12 x$ term.
For the constant terms: $-4 - 9 = -13$

STEP 9

Write the final simplified polynomial in standard form, which is in descending order of the degrees of the terms.
$-2 x^{2} - 12 x - 13$
This is the result when $(2 x+3)^{2}$ is subtracted from $2 x^{2}-4$ .

Was this helpful?

Simplify $(2 x+3)^{2}$ and subtract it from $2 x^{2}-4$ to get the result in standard polynomial form.

STEP 1

STEP 2

First, we need to expand the expression $(2 x+3)^{2}$ .
$(2 x+3)^{2} = (2 x+3)(2 x+3)$

STEP 3

Now, use the distributive property (FOIL method) to expand the product.
$(2 x+3)(2 x+3) = 2 x \cdot 2 x + 2 x \cdot 3 + 3 \cdot 2 x + 3 \cdot 3$

STEP 4

Perform the multiplication for each term.
$2 x \cdot 2 x = 4 x^{2}$ $2 x \cdot 3 = 6 x$ $3 \cdot 2 x = 6 x$ $3 \cdot 3 = 9$

STEP 5

Combine the like terms from the expansion.
$4 x^{2} + 6 x + 6 x + 9 = 4 x^{2} + 12 x + 9$

STEP 6

Now, subtract the expanded expression from the first expression given.
$(2 x^{2}-4) - (4 x^{2} + 12 x + 9)$

STEP 7

Distribute the subtraction across the terms in the second expression.
$2 x^{2}-4 - 4 x^{2} - 12 x - 9$

STEP 8

Combine like terms by subtracting the coefficients of the terms with the same degree.
For the $x^{2}$ terms: $2 x^{2} - 4 x^{2} = -2 x^{2}$
For the $x$ terms: Since there is no $x$ term in the first expression, we only have the $-12 x$ term.
For the constant terms: $-4 - 9 = -13$

STEP 9

Write the final simplified polynomial in standard form, which is in descending order of the degrees of the terms.
$-2 x^{2} - 12 x - 13$
This is the result when $(2 x+3)^{2}$ is subtracted from $2 x^{2}-4$ .

Start learning now

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

Simplify (2x+3)2(2 x+3)^{2}(2x+3)2 and subtract it from 2x2−42 x^{2}-42x2−4 to get the result in standard polynomial form.

STEP 1

STEP 2

First, we need to expand the expression (2x+3)2(2 x+3)^{2}(2x+3)2.(2x+3)2=(2x+3)(2x+3) (2 x+3)^{2} = (2 x+3)(2 x+3) (2x+3)2=(2x+3)(2x+3)

STEP 3

Now, use the distributive property (FOIL method) to expand the product.(2x+3)(2x+3)=2x⋅2x+2x⋅3+3⋅2x+3⋅3 (2 x+3)(2 x+3) = 2 x \cdot 2 x + 2 x \cdot 3 + 3 \cdot 2 x + 3 \cdot 3 (2x+3)(2x+3)=2x⋅2x+2x⋅3+3⋅2x+3⋅3

STEP 4

Perform the multiplication for each term.2x⋅2x=4x2 2 x \cdot 2 x = 4 x^{2} 2x⋅2x=4x2 2x⋅3=6x 2 x \cdot 3 = 6 x 2x⋅3=6x 3⋅2x=6x 3 \cdot 2 x = 6 x 3⋅2x=6x 3⋅3=9 3 \cdot 3 = 9 3⋅3=9

STEP 5

Combine the like terms from the expansion.4x2+6x+6x+9=4x2+12x+9 4 x^{2} + 6 x + 6 x + 9 = 4 x^{2} + 12 x + 9 4x2+6x+6x+9=4x2+12x+9

STEP 6

Now, subtract the expanded expression from the first expression given.(2x2−4)−(4x2+12x+9) (2 x^{2}-4) - (4 x^{2} + 12 x + 9) (2x2−4)−(4x2+12x+9)

STEP 7

Distribute the subtraction across the terms in the second expression.2x2−4−4x2−12x−9 2 x^{2}-4 - 4 x^{2} - 12 x - 9 2x2−4−4x2−12x−9

STEP 8

STEP 9

Write the final simplified polynomial in standard form, which is in descending order of the degrees of the terms.−2x2−12x−13 -2 x^{2} - 12 x - 13 −2x2−12x−13This is the result when (2x+3)2(2 x+3)^{2}(2x+3)2 is subtracted from 2x2−42 x^{2}-42x2−4.

Start learning now

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

Simplify $(2 x+3)^{2}$ and subtract it from $2 x^{2}-4$ to get the result in standard polynomial form.

First, we need to expand the expression $(2 x+3)^{2}$ .
$(2 x+3)^{2} = (2 x+3)(2 x+3)$

Now, use the distributive property (FOIL method) to expand the product.
$(2 x+3)(2 x+3) = 2 x \cdot 2 x + 2 x \cdot 3 + 3 \cdot 2 x + 3 \cdot 3$

Perform the multiplication for each term.
$2 x \cdot 2 x = 4 x^{2}$ $2 x \cdot 3 = 6 x$ $3 \cdot 2 x = 6 x$ $3 \cdot 3 = 9$

Combine the like terms from the expansion.
$4 x^{2} + 6 x + 6 x + 9 = 4 x^{2} + 12 x + 9$

Now, subtract the expanded expression from the first expression given.
$(2 x^{2}-4) - (4 x^{2} + 12 x + 9)$

Distribute the subtraction across the terms in the second expression.
$2 x^{2}-4 - 4 x^{2} - 12 x - 9$

Write the final simplified polynomial in standard form, which is in descending order of the degrees of the terms.
$-2 x^{2} - 12 x - 13$
This is the result when $(2 x+3)^{2}$ is subtracted from $2 x^{2}-4$ .