Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

Calculate the expression:

[tex](m+n+p)^2-(m-n+p)^2[/tex]


Sagot :

Sure, let's solve the expression [tex]\((m+n+p)^2 - (m-n+p)^2\)[/tex] step-by-step.

### Step 1: Understanding the Expression

We need to evaluate the difference of the squares of two expressions: [tex]\(a = (m+n+p)\)[/tex] and [tex]\(b = (m-n+p)\)[/tex].

### Step 2: Using the Difference of Squares Formula

Recall that the difference of squares formula is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]

Here, our [tex]\(a\)[/tex] is [tex]\((m+n+p)\)[/tex] and our [tex]\(b\)[/tex] is [tex]\((m-n+p)\)[/tex].

### Step 3: Calculate [tex]\(a - b\)[/tex]

Let's substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into [tex]\(a - b\)[/tex]:
[tex]\[ a - b = (m+n+p) - (m-n+p) \][/tex]
Simplifying this:
[tex]\[ a - b = m + n + p - m + n - p \][/tex]
The [tex]\(m\)[/tex] and [tex]\(-m\)[/tex] cancel each other out, and the [tex]\(p\)[/tex] and [tex]\(-p\)[/tex] cancel each other out:
[tex]\[ a - b = 2n \][/tex]

### Step 4: Calculate [tex]\(a + b\)[/tex]

Now let’s substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into [tex]\(a + b\)[/tex]:
[tex]\[ a + b = (m+n+p) + (m-n+p) \][/tex]
Simplifying this:
[tex]\[ a + b = m + n + p + m - n + p \][/tex]
The [tex]\(+n\)[/tex] and the [tex]\(-n\)[/tex] cancel each other out:
[tex]\[ a + b = 2m + 2p \][/tex]

### Step 5: Multiply [tex]\(a - b\)[/tex] and [tex]\(a + b\)[/tex]

From the previous steps, we have:
[tex]\[ a - b = 2n \][/tex]
[tex]\[ a + b = 2m + 2p \][/tex]

Now, multiply these two results:
[tex]\[ (a - b)(a + b) = (2n)(2m + 2p) \][/tex]
Distribute [tex]\(2n\)[/tex] across [tex]\(2m + 2p\)[/tex]:
[tex]\[ (2n)(2m) + (2n)(2p) = 4mn + 4np \][/tex]

### Step 6: Final Answer

Thus, [tex]\((m+n+p)^2 - (m-n+p)^2\)[/tex] evaluates to:
[tex]\[ 4mn + 4np \][/tex]

Considering the given specific values, [tex]\(m = 1\)[/tex], [tex]\(n = 2\)[/tex], and [tex]\(p = 3\)[/tex], the numerical result of the expression is:
[tex]\[ 32 \][/tex]
So, the solution is [tex]\(4mn + 4np = 32\)[/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.