Solve for Missing Terms in 23x(? + ?) = 46x² + 23

Q: Why can't the missing terms be 2x and 1?

If the terms were 2x and 1, then $ 23x(2x + 1) = 46x^2 + 23x $, but we need $ 46x^2 + 23 $. The constant term is different!

Q: How do I know to use 1/x as one of the terms?

Look at the constant term! We have $ 23 $ on the right side. When $ 23x $ multiplies by $ \frac{1}{x} $, we get 23, which matches perfectly.

Q: Can I factor out 23 first instead of 23x?

Yes! You can factor $ 46x^2 + 23 = 23(2x^2 + 1) $, but then you'd need $ 23x(? + ?) = 23(2x^2 + 1) $, making it harder to find the missing terms.

Q: Is 1/x a valid algebraic expression?

Absolutely! $ \frac{1}{x} $ is a rational expression. Just remember it's undefined when x = 0, but it's perfectly valid for all other values of x.

Q: How can I check my answer quickly?

Distribute: $ 23x \cdot 2x = 46x^2 $ and $ 23x \cdot \frac{1}{x} = 23 $. Adding gives $ 46x^2 + 23 $ ✓

Distributive Property with Rational Expressions

Fill in the missing values:

$23x(?+?)=46x^2+23$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing values

00:07 Break down 46 into factors 23 and 2

00:10 Break down the square into multiplications

00:25 Add an appropriate whole fraction for the common factor

00:31 Mark the common factors

00:49 Take out the common factors from the parentheses

00:59 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:

$23x(?+?)=46x^2+23$

Step-by-step solution

To solve this problem, let's consider how the equation is structured:

The given equation is $23x(?+?) = 46x^2 + 23$ .

On the left-hand side, we have $23x \times (?+?)$ and on the right-hand side, we observe it's structured as a multiplication between two terms plus a constant.

Re-examine the right-hand side, $46x^2 + 23$ . This can be interpreted as $(2x) \cdot (23x) + 23$ .

The expression can be rearranged as:
$23 \cdot (2x \cdot x + 1)$ .

Therefore, the original equation takes the form:
$23x(2x + \frac{1}{x})$ , also known as multiplying through distribution yields,
$23x \cdot 2x + 23x \cdot \frac{1}{x} = 46x^2 + 23$ ,
which matches perfectly with $46x^2 + 23$ .

By comparing the elements, we find that the missing parts are:

The first missing part is $2x$ .
The second missing part is $\frac{1}{x}$ .

Hence, the values for the question marks are $2x$ and $\frac{1}{x}$ .

Therefore, the correct answer is:

$2x,\frac{1}{x}$

Final Answer

$2x,\frac{1}{x}$

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Factor out common terms to identify missing factors
Technique: Rewrite $46x^2 + 23$ as $23x(2x + \frac{1}{x})$
Check: Verify $23x \cdot 2x + 23x \cdot \frac{1}{x} = 46x^2 + 23$ ✓

Common Mistakes

Avoid these frequent errors

Assuming both missing terms must be polynomials
Don't force both terms to be polynomials like 2x and 1 = wrong factorization! This ignores that rational expressions can have fractional terms. Always consider that one factor might be $\frac{1}{x}$ when the constant term appears.

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 the missing terms be 2x and 1?

If the terms were 2x and 1, then $23x(2x + 1) = 46x^2 + 23x$ , but we need $46x^2 + 23$ . The constant term is different!

How do I know to use 1/x as one of the terms?

Look at the constant term! We have $23$ on the right side. When $23x$ multiplies by $\frac{1}{x}$ , we get 23, which matches perfectly.

Can I factor out 23 first instead of 23x?

Yes! You can factor $46x^2 + 23 = 23(2x^2 + 1)$ , but then you'd need $23x(? + ?) = 23(2x^2 + 1)$ , making it harder to find the missing terms.

Is 1/x a valid algebraic expression?

Absolutely! $\frac{1}{x}$ is a rational expression. Just remember it's undefined when x = 0, but it's perfectly valid for all other values of x.

How can I check my answer quickly?

Distribute: $23x \cdot 2x = 46x^2$ and $23x \cdot \frac{1}{x} = 23$ . Adding gives $46x^2 + 23$ ✓

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