Expanding (x+7)(x-2): Find the Missing Term in Quadratic Expression

Q: Why do I need to multiply every term by every other term?

The distribution law requires each term in the first parentheses to multiply with each term in the second. This ensures you capture all possible combinations and don't miss any terms in your final expression.

Q: How do I keep track of all the multiplications?

Use the FOIL method: First terms (x·x), Outer terms (x·(-2)), Inner terms (7·x), and Last terms (7·(-2)). This systematic approach prevents missed terms.

Q: What if I get confused with the negative signs?

Remember that the sign belongs to the term! So (x-2) means (x+(-2)). When multiplying: positive × negative = negative, and negative × negative = positive.

Q: How do I combine like terms correctly?

Look for terms with the same variable and exponent. Here, -2x and +7x are like terms because both have x¹. Add their coefficients: -2 + 7 = 5, giving you 5x.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing

00:03 We will use the shortened multiplication formulas

00:16 We will match the numbers to the appropriate variables

00:24 We will substitute accordingly to find the missing value

00:31 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

Complete the following equation:

_{$(x+7)(x-2)=x^2+\textcolor{red}{☐}x-14$}

2

Step-by-step solution

Simplify the given expression on the left side:

$(x+7)(x-2)$ Open the parentheses by using the expanded distribution law:

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

Note that in the formula template for the distribution law mentioned, the operation between terms inside the parentheses is automatically addition. Likewise remember that the sign preceding the term is an integral part of it. Proceed to apply the rules of sign multiplication allowing us to present any expression inside of parentheses. The parentheses can be opened using the above formula, first, as an expression where addition operation exists between all terms (if needed),

Therefore, we'll present each of the expressions inside of the parentheses in the multiplication on the left side as an expression with addition operation:

$(x+7)(x-2)=x^2+\textcolor{purple}{\boxed{?}}x-14 \\ \downarrow\\ \big(x+(+7)\big)\big(x+(-2)\big)=x^2+\textcolor{purple}{\boxed{?}}x-14 \\$ Let's once again write the expanded distribution law mentioned earlier:

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

And apply it to our problem:

$\big (\textcolor{red}{x}+\textcolor{blue}{(+7)}\big)\big(x+(-2)\big)=x^2+\textcolor{purple}{\boxed{?}}x-14 \\ \downarrow\\ \textcolor{red}{x}\cdot x +\textcolor{red}{x}\cdot (-2)+\textcolor{blue}{(+7)}\cdot x +\textcolor{blue}{(+7)}\cdot (-2)=x^2+\textcolor{purple}{\boxed{?}}x-14 \\$ Proceed to apply the multiplication sign rules. Remember that multiplying terms with identical signs yields a positive result, and multiplying terms with different signs yields a negative result. In the next step combine like terms in the expression obtained on the left side:

$\textcolor{red}{x}\cdot x +\textcolor{red}{x}\cdot (-2)+\textcolor{blue}{(+7)}\cdot x +\textcolor{blue}{(+7)}\cdot (-2)=x^2+\textcolor{purple}{\boxed{?}}x-14 \\ \downarrow\\ x^2-2x+7x-14=x^2+\textcolor{purple}{\boxed{?}}x-14 \\ x^2+5x-14=x^2+\textcolor{purple}{\boxed{?}}x-14$

Therefore the missing expression is the number 5,

Meaning - the correct answer is A.

3

Final Answer

5

Key Points to Remember

Essential concepts to master this topic

Distribution Law: Multiply each term in first parentheses by each term in second
Technique: For (x+7)(x-2): x·x + x·(-2) + 7·x + 7·(-2) = x² + 5x - 14
Check: Combine like terms carefully: -2x + 7x = 5x coefficient ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying terms across parentheses
Don't add 7 and -2 to get 5 directly = wrong process! This skips the essential multiplication step. Always multiply each term in the first parentheses by each term in the second parentheses, then combine like terms.

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

Why do I need to multiply every term by every other term?

+

The distribution law requires each term in the first parentheses to multiply with each term in the second. This ensures you capture all possible combinations and don't miss any terms in your final expression.

How do I keep track of all the multiplications?

+

Use the FOIL method: First terms (x·x), Outer terms (x·(-2)), Inner terms (7·x), and Last terms (7·(-2)). This systematic approach prevents missed terms.

What if I get confused with the negative signs?

+

Remember that the sign belongs to the term! So (x-2) means (x+(-2)). When multiplying: positive × negative = negative, and negative × negative = positive.

How do I combine like terms correctly?

+

Look for terms with the same variable and exponent. Here, -2x and +7x are like terms because both have x¹. Add their coefficients: -2 + 7 = 5, giving you 5x.

Can I check my answer without expanding everything again?

+

Yes! Substitute a simple value like x = 1 into both sides. Left side: (1+7)(1-2) = 8(-1) = -8. Right side: 1² + 5(1) - 14 = 1 + 5 - 14 = -8. They match! ✓

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Expanding (x+7)(x-2): Find the Missing Term in Quadratic Expression

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