Question Find the product of the sum and difference of two terms: $(7x^2 + 3)(7x^2 - 3)$ .

Studdy Solution

STEP 1

Assumptions
1. We are given two expressions: $(7x^2 + 3)$ and $(7x^2 - 3)$ .
2. We need to find the product of these two expressions.

STEP 2

The product of the sum and difference of two terms can be calculated using the formula for the difference of squares, which is given by:
$(a + b)(a - b) = a^2 - b^2$

STEP 3

Identify the terms $a$ and $b$ in the given expressions. Here, $a = 7x^2$ and $b = 3$ .

STEP 4

Apply the difference of squares formula to the given expressions.
$(7x^2 + 3)(7x^2 - 3) = (7x^2)^2 - (3)^2$

STEP 5

Calculate the square of $7x^2$ .
$(7x^2)^2 = 49x^4$

STEP 6

Calculate the square of $3$ .
$(3)^2 = 9$

STEP 7

Substitute the squared values back into the equation.
$49x^4 - 9$

STEP 8

The final product of the sum and difference of the two terms is:
$\left(7 x^{2}+3\right)\left(7 x^{2}-3\right) = 49x^4 - 9$

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Question Find the product of the sum and difference of two terms: $(7x^2 + 3)(7x^2 - 3)$ .

Studdy Solution

STEP 1

Assumptions
1. We are given two expressions: $(7x^2 + 3)$ and $(7x^2 - 3)$ .
2. We need to find the product of these two expressions.

STEP 2

The product of the sum and difference of two terms can be calculated using the formula for the difference of squares, which is given by:
$(a + b)(a - b) = a^2 - b^2$

STEP 3

Identify the terms $a$ and $b$ in the given expressions. Here, $a = 7x^2$ and $b = 3$ .

STEP 4

Apply the difference of squares formula to the given expressions.
$(7x^2 + 3)(7x^2 - 3) = (7x^2)^2 - (3)^2$

STEP 5

Calculate the square of $7x^2$ .
$(7x^2)^2 = 49x^4$

STEP 6

Calculate the square of $3$ .
$(3)^2 = 9$

STEP 7

Substitute the squared values back into the equation.
$49x^4 - 9$

STEP 8

The final product of the sum and difference of the two terms is:
$\left(7 x^{2}+3\right)\left(7 x^{2}-3\right) = 49x^4 - 9$

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Question Find the product of the sum and difference of two terms: (7x2+3)(7x2−3)(7x^2 + 3)(7x^2 - 3)(7x2+3)(7x2−3).

Studdy Solution

STEP 1

Assumptions1. We are given two expressions: (7x2+3)(7x^2 + 3)(7x2+3) and (7x2−3)(7x^2 - 3)(7x2−3).2. We need to find the product of these two expressions.

STEP 2

The product of the sum and difference of two terms can be calculated using the formula for the difference of squares, which is given by:(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2(a+b)(a−b)=a2−b2

STEP 3

Identify the terms aaa and bbb in the given expressions. Here, a=7x2a = 7x^2a=7x2 and b=3b = 3b=3.

STEP 4

Apply the difference of squares formula to the given expressions.(7x2+3)(7x2−3)=(7x2)2−(3)2(7x^2 + 3)(7x^2 - 3) = (7x^2)^2 - (3)^2(7x2+3)(7x2−3)=(7x2)2−(3)2

STEP 5

Calculate the square of 7x27x^27x2.(7x2)2=49x4(7x^2)^2 = 49x^4(7x2)2=49x4

STEP 6

Calculate the square of 333.(3)2=9(3)^2 = 9(3)2=9

STEP 7

Substitute the squared values back into the equation.49x4−949x^4 - 949x4−9

STEP 8

The final product of the sum and difference of the two terms is:(7x2+3)(7x2−3)=49x4−9\left(7 x^{2}+3\right)\left(7 x^{2}-3\right) = 49x^4 - 9(7x2+3)(7x2−3)=49x4−9

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Question Find the product of the sum and difference of two terms: $(7x^2 + 3)(7x^2 - 3)$ .

Assumptions
1. We are given two expressions: $(7x^2 + 3)$ and $(7x^2 - 3)$ .
2. We need to find the product of these two expressions.

The product of the sum and difference of two terms can be calculated using the formula for the difference of squares, which is given by:
$(a + b)(a - b) = a^2 - b^2$

Identify the terms $a$ and $b$ in the given expressions. Here, $a = 7x^2$ and $b = 3$ .

Apply the difference of squares formula to the given expressions.
$(7x^2 + 3)(7x^2 - 3) = (7x^2)^2 - (3)^2$

Calculate the square of $7x^2$ .
$(7x^2)^2 = 49x^4$

Calculate the square of $3$ .
$(3)^2 = 9$

Substitute the squared values back into the equation.
$49x^4 - 9$

The final product of the sum and difference of the two terms is:
$\left(7 x^{2}+3\right)\left(7 x^{2}-3\right) = 49x^4 - 9$