Solve the Equation: 32(? + ?) = 8 + 2a - Finding Missing Values

Q: Why can't I just divide both sides by 32?

Dividing gives you $ ? + ? = \frac{8 + 2a}{32} $, but you need two separate values that add up to this result. You must factor each term individually!

Q: Can the missing values be negative?

Yes! Missing values can be any real numbers (positive, negative, or zero) as long as when multiplied by 32 and added together, they equal $ 8 + 2a $.

Q: What if there were different numbers instead of 32, 8, and 2a?

The same method works! Factor out the coefficient from each term on the right side. If you had $ 20(? + ?) = 15 + 4b $, you'd get $ \frac{15}{20} + \frac{4b}{20} $.

Q: How can I check my answer quickly?

Substitute back: $ 32\left(\frac{1}{4} + \frac{1}{16}a\right) = 32 \cdot \frac{1}{4} + 32 \cdot \frac{1}{16}a = 8 + 2a $ ✓

Factoring Expressions with Variable Coefficients

Fill in the missing values:

$32(?+?)=8+2a$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing values

00:04 Multiply by the appropriate whole fraction

00:21 Raise the products to the numerator

00:25 Factor 32 into factors 8 and 4

00:32 Factor 32 into factors 16 and 2

00:35 Reduce what is possible

00:49 Mark the common factors

00:52 Take out the common factors from the parentheses

01:02 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the missing values:

$32(?+?)=8+2a$

Step-by-step solution

To solve the equation $32(?+?)=8+2a$ , follow these steps:

First, observe the equation: $32(? + ?) = 8 + 2a$ .
Factor out $32$ on the left side: it should already involve distribution.
On the right side, we see $8 + 2a$ can potentially be related to $32$ : $8 = \frac{1}{4} \cdot 32$ and $2a = \frac{1}{16}a \cdot 32$ .
Therefore, express the equation as:

32\left(\frac{1}{4} + \frac{1}{16}a\right) = 8 + 2a

This demonstrates that the missing values to satisfy the equation's balance, by distribution properties, are:

\frac{1}{4} \text{ and } \frac{1}{16}a

Thus, the completed form of the right side of the equation with respect to $32$ is:

Hence, the solution is: $\frac{1}{4}, \frac{1}{16}a$ .

Final Answer

$\frac{1}{4},\frac{1}{16}a$

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Factor out common terms to find missing values
Technique: Express 8 as $\frac{1}{4} \cdot 32$ and 2a as $\frac{1}{16}a \cdot 32$
Check: Verify $32\left(\frac{1}{4} + \frac{1}{16}a\right) = 8 + 2a$ by distributing ✓

Common Mistakes

Avoid these frequent errors

Trying to solve for a single unknown variable
Don't treat this as $32x = 8 + 2a$ and solve for x = wrong approach! This creates confusion because you're looking for two separate values. Always recognize you need to factor out the coefficient from each term on the right side.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

Why can't I just divide both sides by 32?

Dividing gives you $? + ? = \frac{8 + 2a}{32}$ , but you need two separate values that add up to this result. You must factor each term individually!

How do I know what to factor out of each term?

Look at each term on the right: 8 and 2a. Ask yourself: "What times 32 gives me 8?" That's $\frac{1}{4}$ . Then: "What times 32 gives me 2a?" That's $\frac{1}{16}a$ .

Can the missing values be negative?

Yes! Missing values can be any real numbers (positive, negative, or zero) as long as when multiplied by 32 and added together, they equal $8 + 2a$ .

What if there were different numbers instead of 32, 8, and 2a?

The same method works! Factor out the coefficient from each term on the right side. If you had $20(? + ?) = 15 + 4b$ , you'd get $\frac{15}{20} + \frac{4b}{20}$ .

How can I check my answer quickly?

Substitute back: $32\left(\frac{1}{4} + \frac{1}{16}a\right) = 32 \cdot \frac{1}{4} + 32 \cdot \frac{1}{16}a = 8 + 2a$ ✓

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