Question

*. completa :
a) $(3 + x)^2 =$
b) $(x + 1)^2 =$
c) $(2x + 5)^2 =$
d) $(3 + y^2)^2 =

Explanation:

To solve these problems, we use the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).

Part a) \((3 + x)^2\)

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

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

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (3 + x)^2&=3^2+2\times3\times x + x^2\\ &= 9 + 6x + x^2 \end{align*}$$

\]

Part b) \((x + 1)^2\)

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

Here, \(a = x\) and \(b = 1\).

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (x + 1)^2&=x^2+2\times x\times1 + 1^2\\ &=x^2 + 2x + 1 \end{align*}$$

\]

Part c) \((2x + 5)^2\)

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

Here, \(a = 2x\) and \(b = 5\).

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (2x + 5)^2&=(2x)^2+2\times(2x)\times5 + 5^2\\ &=4x^2+20x + 25 \end{align*}$$

\]

Part d) \((3 + y^2)^2\)

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

Here, \(a = 3\) and \(b = y^2\).

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (3 + y^2)^2&=3^2+2\times3\times y^2+(y^2)^2\\ &=9 + 6y^2 + y^4 \end{align*}$$

\]

Respuestas:

a) \(\boldsymbol{9 + 6x + x^2}\)
b) \(\boldsymbol{x^2 + 2x + 1}\)
c) \(\boldsymbol{4x^2 + 20x + 25}\)
d) \(\boldsymbol{9 + 6y^2 + y^4}\)

Answer:

To solve these problems, we use the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).

Part a) \((3 + x)^2\)

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

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

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (3 + x)^2&=3^2+2\times3\times x + x^2\\ &= 9 + 6x + x^2 \end{align*}$$

\]

Part b) \((x + 1)^2\)

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

Here, \(a = x\) and \(b = 1\).

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (x + 1)^2&=x^2+2\times x\times1 + 1^2\\ &=x^2 + 2x + 1 \end{align*}$$

\]

Part c) \((2x + 5)^2\)

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

Here, \(a = 2x\) and \(b = 5\).

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (2x + 5)^2&=(2x)^2+2\times(2x)\times5 + 5^2\\ &=4x^2+20x + 25 \end{align*}$$

\]

Part d) \((3 + y^2)^2\)

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

Here, \(a = 3\) and \(b = y^2\).

Step 2: Apply the binomial square formula

\[

$$\begin{align*} (3 + y^2)^2&=3^2+2\times3\times y^2+(y^2)^2\\ &=9 + 6y^2 + y^4 \end{align*}$$

\]

Respuestas:

a) \(\boldsymbol{9 + 6x + x^2}\)
b) \(\boldsymbol{x^2 + 2x + 1}\)
c) \(\boldsymbol{4x^2 + 20x + 25}\)
d) \(\boldsymbol{9 + 6y^2 + y^4}\)