Difference of squares: Complete the missing numbers

To solve this problem, we'll follow these steps:

• Identify the given expression as a difference of squares.
• Apply the formula for finding the missing term in $(x+a)(x-a) = x^2 - a^2$.
• Determine the values of $a$ to fill in the blanks.

Now, let's work through each step:
Step 1: The expression given is $(x+\_—)\cdot(x-\_—) = x^2-121$. Recognize that $x^2 - 121$ is a difference of squares.
Step 2: We know from the difference of squares formula that $a^2 = 121$.
Step 3: Solve for $a$ by taking the square root of both sides: $a = \sqrt{121} = 11$.

This means the expression becomes: $(x+11)(x-11) = x^2 - 121$.

Therefore, the missing element is $11$.

To solve this problem, let's use the difference of squares formula, which is $(a + b)(a - b) = a^2 - b^2$. Given the equation $(_ + 3)(_- 3) = x^2 - 9$, we can compare it to the formula:

• $a^2 = x^2$ implies $a = x$.
• $b^2 = 9$ implies $b = 3$.

This means the expression $(_ + 3)(_- 3)$ should represent $(x + 3)(x - 3)$, satisfying the equation through the difference of squares formula.

Thus, the missing element to obtain a correct expression is $x$.

To solve this problem, we need to recognize the expression $x^2 - 64$ as a difference of squares.

The difference of squares formula states: $a^2 - b^2 = (a - b)(a + b)$.

In this problem, we identify that:

• $a = x$

• $b^2 = 64$, which means $b = \sqrt{64} = 8$

Therefore, applying the formula gives us:

$x^2 - 64 = (x - 8)(x + 8)$

This indicates that the missing element in the expression $(x - \_)(\_ + x)$ is $8$.

Thus, the correct answer to fill in the missing element is $\boxed{8}$, corresponding to choice 4.

To solve this problem, we'll follow these steps:

• Step 1: Recognize that $49 = 7^2$.
• Step 2: Apply the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$.
• Step 3: Compare the equation to the form and determine the blanks as $7$.

Now, let's work through each step:

Step 1: The given expression is $x^2 - 49$. Observe that 49 is a perfect square, written as $7^2$.

Step 2: According to the difference of squares formula, $x^2 - 49$ can be rewritten as $x^2 - 7^2$, which equals $(x - 7)(x + 7)$.

Step 3: Plugging in our values, we know the expression matches the form $(x - \_)\cdot(x + \_)$, with $7$ being the missing number.

Therefore, the solution to the problem is 7, which corresponds to choice 2.

To solve the problem, we need to express $x^2 - 6$ in the form of $(x-a)\cdot(x+a)$ because this represents the difference of squares, which is expressed as $(a-b)(a+b) = a^2 - b^2$.

We are given $x^2 - 6$. Compare this to the formula $x^2 - b^2$, it suggests that $b^2 = 6$.

The next step is to solve for $b$ by taking the square root of both sides:

$b^2 = 6 \Rightarrow b = \sqrt{6}$.

Thus, the missing number that completes the expression is $\sqrt{6}$.

Therefore, the solution to the problem is $\sqrt{6}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given expression and recognize the form.
• Step 2: Apply the difference of squares formula.
• Step 3: Determine the missing element.
• Step 4: Verify the solution against the possible choices.

Now, let's work through each step:
Step 1: The given expression is $x^2 - 36$. This resembles a difference of squares, which is $a^2 - b^2$.
Step 2: Recognize that $x^2$ represents $a^2$ and $36$ represents $b^2$.
Step 3: Find $b$ such that $b^2 = 36$. This gives $b = 6$ because $6^2 = 36$.
Step 4: The difference of squares formula states $a^2 - b^2 = (a - b)(a + b)$. So we rewrite $x^2 - 36$ as $(x - 6)(x + 6)$.

Therefore, the missing element that makes the expression true is $6$.

To solve this problem, we'll follow these steps:

• Step 1: Expand the expression on the right side of the equation.
• Step 2: Compare it with the left-hand side equation and find the missing number.

Now, let's work through each step:
Step 1: Expand the expression $2(x-4)(x+4)$. Using the difference of squares, this becomes:

Step 2: Compare it with the original left side $2x^2 - \_ = 2x^2 - 32$.

The missing number must be 32 so that both sides of the equation are equal.

Therefore, the solution to the problem is $\textbf{32}$.