Expand (x+7)(x-9): Find the Missing Coefficient in x²+□x-63

Q: How do I remember which terms to multiply together?

Use FOIL: First × First, Outer × Outer, Inner × Inner, Last × Last. For $ (x+7)(x-9) $, that's x·x, x·(-9), 7·x, 7·(-9).

Q: Why is the coefficient negative when I get -2?

Because $ -9x + 7x = -2x $! The x·(-9) gives you -9x, and 7·x gives you +7x. When you combine like terms, -9 + 7 = -2.

Q: How can I check if -2 is the right coefficient?

Substitute back! If the coefficient is -2, then $ x^2 - 2x - 63 $ should equal your original expansion. Try with a test value like x = 1.

Q: What if I forget the negative sign?

Pay close attention to signs when distributing! $ (x-9) $ means $ (x + (-9)) $, so multiplying by anything gives you a negative result.

Q: Do I always get the constant term by multiplying the last numbers?

Yes! The constant term always comes from multiplying the constant terms together. Here: $ 7 \times (-9) = -63 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing

00:04 We will use shortened multiplication formulas

00:13 We will match the numbers to the appropriate variables

00:22 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:

_{$(x+7)(x-9)=x^2+\textcolor{red}{☐}x-63$}

2

Step-by-step solution

Simplify the given expression on the left side:

$(x+7)(x-9)$

Proceed to open the parentheses 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, we take by default that the operation between terms inside of the parentheses is addition. It's vital to remember that the sign preceding the term is an integral part of it. By applying the rules of sign multiplication we can present any expression inside of the parentheses. Open the parentheses using the above formula, first, as an expression where addition occurs between all terms (if needed),

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

$(x+7)(x-9)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ \downarrow\\ \big(x+(+7)\big)\big(x+(-9)\big)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\$ Let's write the expanded distribution law mentioned earlier:

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

Proceed to apply it to our problem:

$\big (\textcolor{red}{x}+\textcolor{blue}{(+7)}\big)\big(x+(-9)\big)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ \downarrow\\ \textcolor{red}{x}\cdot x +\textcolor{red}{x}\cdot (-9)+\textcolor{blue}{(+7)}\cdot x +\textcolor{blue}{(+7)}\cdot (-9)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\$ Continue 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:

$\textcolor{red}{x}\cdot x +\textcolor{red}{x}\cdot (-9)+\textcolor{blue}{(+7)}\cdot x +\textcolor{blue}{(+7)}\cdot (-9)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ \downarrow\\ x^2-9x+7x-63=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ x^2-2x-63=x^2+\textcolor{purple}{\boxed{?}}x-63$

Therefore the missing expression is the number $-2$ ,

That is - the correct answer is a'.

3

Final Answer

2-

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Multiply each term in first parentheses by each in second
Technique: $x \cdot (-9) + 7 \cdot x = -9x + 7x = -2x$
Check: Expand completely and verify coefficients match given form ✓

Common Mistakes

Avoid these frequent errors

Adding instead of distributing properly
Don't just add the numbers inside parentheses like (x+7)(x-9) = x + 7 + x - 9 = 2x - 2! This ignores the multiplication between binomials and gives completely wrong results. Always distribute each term from the first binomial to each term in the second binomial.

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

How do I remember which terms to multiply together?

+

Use FOIL: First × First, Outer × Outer, Inner × Inner, Last × Last. For $(x+7)(x-9)$ , that's x·x, x·(-9), 7·x, 7·(-9).

Why is the coefficient negative when I get -2?

+

Because $-9x + 7x = -2x$ ! The x·(-9) gives you -9x, and 7·x gives you +7x. When you combine like terms, -9 + 7 = -2.

How can I check if -2 is the right coefficient?

+

Substitute back! If the coefficient is -2, then $x^2 - 2x - 63$ should equal your original expansion. Try with a test value like x = 1.

What if I forget the negative sign?

+

Pay close attention to signs when distributing! $(x-9)$ means $(x + (-9))$ , so multiplying by anything gives you a negative result.

Do I always get the constant term by multiplying the last numbers?

+

Yes! The constant term always comes from multiplying the constant terms together. Here: $7 \times (-9) = -63$ .

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Expand (x+7)(x-9): Find the Missing Coefficient in x²+□x-63

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