Question

Yes or No? If No, give a reason.

(a) Is the expression${\left(x+5\right)}^2$equal to$x^2+25$?

(b) When you expand${\left(x+a\right)}^2$, where$a\ne 0$, do you get three terms?

(c) Is the expression$\left(x+5\right)\left(x-5\right)$ equal to$x^2-25$?

(d) When you expand$\left(x+a\right)\left(x-a\right)$, where$a\ne 0$, do you get two terms?

Short answer

1. No,the expression ${\left(x+5\right)}^2$ is not equal to $x^2+25$. ${\left(x+5\right)}^2=x^2+10x+25$
2. Yes,when we expand ${\left(x+a\right)}^2$where $a\ne 0$, we get three terms.
3. Yes,the expression $\left(x+5\right)\left(x-5\right)$ is equal to $x^2-25$
4. Yes,when we expand$\left(x+a\right)\left(x-a\right)$ where ,$a\ne 0$ we get two terms.

Step-by-step solution

Part a. Step 1. Apply the “square of the sum” formula.

If A and B are any real numbers or algebraic expressions, then

${\left(A+B\right)}^2=A^2+2AB+B^2$

Part a. Step 2. Analyze the given expression.

Given${\left(x+5\right)}^2$

Here

$A=x,B=5$

Part a. Step 3. Solve the expression.

Substitute the values in the square of the sum formula to get:

$\begin{array}{l}{\left(x+5\right)}^2=x^2+2\cdot x\cdot 5+5^2\\ \Rightarrow {\left(x+5\right)}^2=x^2+10x+25\end{array}$

So, the expression${\left(x+5\right)}^2$ is not equal to$x^2+25$

Part b. Step 1. Apply the “square of the sum” formula.

If A and B are any real numbers or algebraic expressions, then

${\left(A+B\right)}^2=A^2+2AB+B^2$

Part b. Step 2. Analyze the given expression.

Given ${\left(x+a\right)}^2$,$a\ne 0$

Here

$A=x,B=a$

Part b. Step 3. Solve the expression.

Substitute the values in the square of the sum formula to get:

$\begin{array}{l}{\left(x+a\right)}^2=x^2+2\cdot x\cdot a+a^2\\ \Rightarrow {\left(x+5\right)}^2=x^2+2ax+a^2\end{array}$

So, when we expand${\left(x+a\right)}^2$ where $a\ne 0$, we get three terms.

Part c. Step 1. Apply the “product of the sum and difference of terms” formula.

If A and B are any real numbers or algebraic expressions, then

$\left(A+B\right)\left(A-B\right)=A^2-B^2$

Part c. Step 2. Analyze the given expression.

Given$\left(x+5\right)\left(x-5\right)$

Here$A=x,B=5$

Part c. Step 3. Solve the expression.

Substitute the values in the product of the sum and difference of terms formula to get:

$\begin{array}{l}\left(x+5\right)\left(x-5\right)=x^2-5^2\\ \Rightarrow \left(x+5\right)\left(x-5\right)=x^2-25\end{array}$

So, the expression$\left(x+5\right)\left(x-5\right)$ is equal to$x^2-25$

Part d. Step 1. Apply the “product of the sum and difference of terms” formula.

If A and B are any real numbers or algebraic expressions, then

$\left(A+B\right)\left(A-B\right)=A^2-B^2$

Part d. Step 2. Analyze the given expression.

Given$\left(x+a\right)\left(x-a\right)$$a\ne 0$

Here

$A=x,B=a$

Part d. Step 3. Solve the expression.

Substitute the values in the product of the sum and difference of terms formula to get:

$\left(x+a\right)\left(x-a\right)=x^2-a^2$

So, when we expand$\left(x+a\right)\left(x-a\right)$ where $a\ne 0$, we get two terms.