Question

el cuadrado de una diferencia:
$(a - b)^2 = a^2 - 2ab + b^2$
$(a - b) \cdot (a - b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2$

• completa:

a) $(x - 3)^2 = $
b) $(7 - y)^2 = $
c) $(5 - 3x)^2 = $
d) $(y^2 - 2)^2 = $

Explanation:

To solve the given problems using the square of a difference formula \((a - b)^2 = a^2 - 2ab + b^2\), we analyze each one:

Problem 2: \((x - 3)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = x\) and \(b = 3\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = x\) and \(b = 3\) into the formula:
\[

$$\begin{align*} (x - 3)^2 &= x^2 - 2 \cdot x \cdot 3 + 3^2 \\ &= x^2 - 6x + 9 \end{align*}$$

\]

Problem 3: \((7 - y)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = 7\) and \(b = y\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = 7\) and \(b = y\) into the formula:
\[

$$\begin{align*} (7 - y)^2 &= 7^2 - 2 \cdot 7 \cdot y + y^2 \\ &= 49 - 14y + y^2 \end{align*}$$

\]

Problem 4: \((5 - 3x)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = 5\) and \(b = 3x\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = 5\) and \(b = 3x\) into the formula:
\[

$$\begin{align*} (5 - 3x)^2 &= 5^2 - 2 \cdot 5 \cdot 3x + (3x)^2 \\ &= 25 - 30x + 9x^2 \end{align*}$$

\]

Problem 5: \((y^2 - 2)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = y^2\) and \(b = 2\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = y^2\) and \(b = 2\) into the formula:
\[

$$\begin{align*} (y^2 - 2)^2 &= (y^2)^2 - 2 \cdot y^2 \cdot 2 + 2^2 \\ &= y^4 - 4y^2 + 4 \end{align*}$$

\]

Respuestas:

• \((x - 3)^2 = \boldsymbol{x^2 - 6x + 9}\)
• \((7 - y)^2 = \boldsymbol{49 - 14y + y^2}\)
• \((5 - 3x)^2 = \boldsymbol{25 - 30x + 9x^2}\)
• \((y^2 - 2)^2 = \boldsymbol{y^4 - 4y^2 + 4}\)

Answer:

To solve the given problems using the square of a difference formula \((a - b)^2 = a^2 - 2ab + b^2\), we analyze each one:

Problem 2: \((x - 3)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = x\) and \(b = 3\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = x\) and \(b = 3\) into the formula:
\[

$$\begin{align*} (x - 3)^2 &= x^2 - 2 \cdot x \cdot 3 + 3^2 \\ &= x^2 - 6x + 9 \end{align*}$$

\]

Problem 3: \((7 - y)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = 7\) and \(b = y\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = 7\) and \(b = y\) into the formula:
\[

$$\begin{align*} (7 - y)^2 &= 7^2 - 2 \cdot 7 \cdot y + y^2 \\ &= 49 - 14y + y^2 \end{align*}$$

\]

Problem 4: \((5 - 3x)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = 5\) and \(b = 3x\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = 5\) and \(b = 3x\) into the formula:
\[

$$\begin{align*} (5 - 3x)^2 &= 5^2 - 2 \cdot 5 \cdot 3x + (3x)^2 \\ &= 25 - 30x + 9x^2 \end{align*}$$

\]

Problem 5: \((y^2 - 2)^2\)

Step 1: Identify \(a\) and \(b\)

Here, \(a = y^2\) and \(b = 2\).

Step 2: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Substitute \(a = y^2\) and \(b = 2\) into the formula:
\[

$$\begin{align*} (y^2 - 2)^2 &= (y^2)^2 - 2 \cdot y^2 \cdot 2 + 2^2 \\ &= y^4 - 4y^2 + 4 \end{align*}$$

\]

Respuestas:

• \((x - 3)^2 = \boldsymbol{x^2 - 6x + 9}\)
• \((7 - y)^2 = \boldsymbol{49 - 14y + y^2}\)
• \((5 - 3x)^2 = \boldsymbol{25 - 30x + 9x^2}\)
• \((y^2 - 2)^2 = \boldsymbol{y^4 - 4y^2 + 4}\)