Match together expressions of equal value$ (x+6)(x+8) $$ (6+x)(8-x) $$ (x+x)(6+8) $a.$ 48+2x-x^2 $b.$ 28x $c.$ x^2+14x+48 $

1-b, 2-a, 3-No suitable expression

Match Equivalent Expressions: (x+6)(x+8) and Related Forms

Q: Why do I get four terms when I multiply two binomials?

Each term in the first parentheses must multiply with every term in the second parentheses. So $ (x+6)(x+8) $ gives you: x·x, x·8, 6·x, and 6·8 = four terms total!

Q: How do I handle subtraction like in (6+x)(8-x)?

Treat the minus sign as part of the term! So $ (8-x) $ becomes $ (8+(-x)) $. Then distribute normally: 6·8 + 6·(-x) + x·8 + x·(-x).

Q: What if I can simplify before distributing?

Great observation! In $ (x+x)(6+8) $, you can combine like terms first: $ 2x \cdot 14 = 28x $. This saves time and reduces errors!

Q: How do I know which expanded form matches which option?

Expand each expression completely, then combine like terms. Compare your final simplified form with the given options. Look for matching coefficients and terms!

Q: Why does the order of terms matter in matching?

The order doesn't matter in addition! $ x^2 + 14x + 48 $ equals $ 48 + 14x + x^2 $. Focus on having the same terms with the same coefficients.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Open parentheses

00:05 Open parentheses properly, multiply each factor by each factor

00:13 Calculate the products

00:17 Collect terms

00:22 This is the simplification for 1, let's continue to 2

00:27 Open parentheses properly, multiply each factor by each factor

00:33 Calculate the products

00:45 Collect terms

00:50 This is the simplification for 2, let's continue to 3

00:55 Calculate each parenthesis and then multiply

00:58 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Match together expressions of equal value

$(x+6)(x+8)$
$(6+x)(8-x)$
$(x+x)(6+8)$
a. $48+2x-x^2$
b. $28x$
c. $x^2+14x+48$

2

Step-by-step solution

Let's simplify the given expressions, open the parentheses using the extended distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

In the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

$(x+6)(x+8) \\ x\cdot x+x\cdot 8+6\cdot x+6\cdot8\\ x^2+8x+6x+48\\ \boxed{x^2+14x+48}\\$
$(6+x)(8-x) \\ \downarrow\\ (6+x)\big(8+(-x)\big) \\ 6\cdot 8+6\cdot (-x)+x\cdot 8+x\cdot(-x)\\ 48-6x+8x-x^2\\ \boxed{48+2x-x^2}\\$
$(x+x)(6+8) \\ 2x\cdot14\\ \boxed{28x}$
In the last expression we simplified above, we first combined like terms in each of the expressions within parentheses, therefore in this case there was no need to use the extended distribution law mentioned at the beginning of the solution to simplify the expression.
After applying the commutative law of addition and multiplication we observe that:
The simplified expression in 1 matches the expression in option C,
The simplified expression in 2 matches the expression in option A,
The simplified expression in 3 matches the expression in option B,

Therefore, the correct answer (among the suggested options) is answer B.

3

Final Answer

1-b, 2-a, 3-b

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Apply (a+b)(c+d) = ac + ad + bc + bd systematically
Technique: For (x+6)(x+8), get x² + 8x + 6x + 48 = x² + 14x + 48
Check: Verify by substituting x=1: (1+6)(1+8) = 63 and 1² + 14(1) + 48 = 63 ✓

Common Mistakes

Avoid these frequent errors

Missing terms when distributing
Don't just multiply the first terms and last terms = incomplete expansion! This gives you only x² + 48 instead of x² + 14x + 48, missing the crucial middle term. Always multiply each term in the first parentheses by each term in the second parentheses.

Practice Quiz

Test your knowledge with interactive questions

$(3+20)\times(12+4)=$

FAQ

Everything you need to know about this question

Why do I get four terms when I multiply two binomials?

+

Each term in the first parentheses must multiply with every term in the second parentheses. So $(x+6)(x+8)$ gives you: x·x, x·8, 6·x, and 6·8 = four terms total!

How do I handle subtraction like in (6+x)(8-x)?

+

Treat the minus sign as part of the term! So $(8-x)$ becomes $(8+(-x))$ . Then distribute normally: 6·8 + 6·(-x) + x·8 + x·(-x).

What if I can simplify before distributing?

+

Great observation! In $(x+x)(6+8)$ , you can combine like terms first: $2x \cdot 14 = 28x$ . This saves time and reduces errors!

How do I know which expanded form matches which option?

+

Expand each expression completely, then combine like terms. Compare your final simplified form with the given options. Look for matching coefficients and terms!

Why does the order of terms matter in matching?

+

The order doesn't matter in addition! $x^2 + 14x + 48$ equals $48 + 14x + x^2$ . Focus on having the same terms with the same coefficients.

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