Factorization - Common Factor: Complete the missing number

To solve the problem $10x(?+?)=20x+30x^2$, we need to determine functions of $x$ such that factorization using the distributive property divides the polynomial as needed.

Step 1: Start by factoring the common term on the left side:

The expression $10x(?+?)$ suggests that $10x$ should factor terms from the polynomial $20x + 30x^2$.

Step 2: Distribute backward:

• The term $20x$ can be rewritten as $10x \times 2$.
• The term $30x^2$ can be rewritten as $10x \times 3x$.

Thus, using factorization, the expression becomes:

$10x(2 + 3x) = 20x + 30x^2$

This completes the expression and verifies the factorization is correct.

Therefore, the missing values are $2, 3x$, corresponding to choice .

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:

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

To solve this problem, we'll follow these steps:

• Step 1: Apply the distributive property to $(4x+8)(b+c)$
• Step 2: Match the expanded terms to $4ax + 8a + 12x + 24$
• Step 3: Solve for $b$ and $c$

Now, let's work through each step:
Step 1: Use the distributive property to expand $(4x+8)(b+c)$. This gives us:

$(4x+8)(b+c) = 4x \cdot b + 8 \cdot b + 4x \cdot c + 8 \cdot c$

Step 2: Equate the expression from Step 1 to $4ax + 8a + 12x + 24$:
$4bx + 8b + 4cx + 8c = 4ax + 8a + 12x + 24$

Separate and equate the coefficients for $x$ and the constant terms:

• For $x$: $4b + 4c = 4a + 12$
• For constants: $8b + 8c = 8a + 24$

Step 3: Solve the resulting system of equations:

Divide each equation by its common factor: - $4b + 4c = 4a + 12$ becomes: $b + c = a + 3$ - $8b + 8c = 8a + 24$ becomes: $b + c = a + 3$

Both equations are identical, thus we only need one further condition to solve completely.

Match assumptions based on simplest composition of terms:
Assume $b = a$ and $c = 3$ to verify this works correctly:

Substituting these into $b + c = a + 3$ gives:
$a + 3 = a + 3$, confirming our choice is consistent.

Thus, the solution to the problem for missing values is $a,3$.

To solve this problem, we need to equate the expression $?(2x+y)$ to $6x+3y$ and solve for the missing factor.

Step 1: Analyze the expression $6x + 3y$.

• Notice that both terms, $6x$ and $3y$, have a common factor of 3.

Thus, we can factor the expression as $3(2x + y)$.

Step 2: Equate the factored form $3(2x + y)$ with the original form $?(2x + y)$.

• If $?(2x + y) = 3(2x + y)$, then the missing factor $?$ must be 3.

Therefore, the missing number that satisfies the equation $?(2x+y)=6x+3y$ is $3$.

To solve this problem, we will expand the left-hand expression and set it equal to the right-hand side.

Let's rewrite the expression: $(x-4)(8y-7)$.

• We begin by expanding the left-hand side using the distributive property:
  $(x-4)(8y-7) = x(8y-7) - 4(8y-7)$.
• Further expand each component:
  $x(8y-7) = 8xy - 7x$ and $-4(8y-7) = -32y + 28$.
• Combine to form:
  $8xy - 7x - 32y + 28$.

We compare this with the right-hand side of the original equation, $8xy - 32y - 7x + 28$.

The expressions match perfectly upon comparative structuring.

Hence, the missing expression in $(x-4)$ confirms accurate factorization.

Thus, the missing values that satisfy the equation are $x-4$.

Therefore, the missing values are $(-4, x)$ or equivalently $x, -4$.

Let's solve the given problem by factoring the expression on the right side and completing the missing part on the left side accordingly,

We will then examine each of the algebraic expressions in both sides of the given equation separately,

On the right side of the equation, the expression:

$10a-90$Let's now examine the expression on the left side of the equation:

$?(18-2a)$Let's note that in this expression there is a factor (unknown) multiplying an expression in parentheses, therefore to understand what this factor is - we'll return to the expression on the right side and factor it using common factor extraction,

In a routine manner - we'll look for first the largest common factor, we'll do this separately, for the numbers and for the letters:

Let's start with the numbers:

In the expression on the right side, there are the numerical coefficients - 90 and 10, let's note that the number 90 is a multiple of the number 10:

$90=9\cdot10$Therefore, the number 10 is the largest common factor for the numbers,

Let's continue and examine the letters:

Let's note that only the left term in the expression on the right side, meaning the term -

$10a$depends on $a$unlike the second term in this expression:

$-90$which does not depend on $a$

$$Therefore, there is no common factor for these two terms (so we'll consider 1 as the common factor for the letters),

Let's summarize:

The largest common factor for numbers and letters together is:

$10\cdot1\\ \downarrow\\ 10$

$$Let's continue then and factor (using common factor extraction) the expression on the right side:

$\textcolor{red}{ 10a} \textcolor{blue}{-90 } \\ \underline{10}\cdot\textcolor{red}{a}+\underline{10}\cdot(\textcolor{blue}{ -9} )\\ \downarrow\\ \underline{10}\cdot(\textcolor{red}{a}\textcolor{blue}{ -9} )$

In the above expression, the operation is explained using colors and markings:

The common factor is highlighted with an underline, and the multipliers inside the parentheses are associated with the terms in the original expression using colors, let's note that we referred in the factoring details above both to the sign of the common factor (in black) that we extracted as a multiplier outside the parentheses and to the signs of the terms in the original expression (in colors), it's not necessary to present this in stages as described above, one can (and should) jump directly to the factored form in the last line, but we must definitely refer to the aforementioned signs, since in each term the sign is an inseparable part of it,

We can easily verify that this factoring is correct easily by opening the parentheses using the distributive property and verifying that indeed we get back the original expression we factored term by term, it's advisable to do this while emphasizing the signs of the terms in the original expression and the sign (always optional) of the common factor.

Let's return to the given problem:

We factored the expression on the right side of the given equation, let's apply this factoring in the equation itself:

$?(18-2a)=10a-90 \\ \downarrow\\ ?(18-2a)=10(a-9)$

Let's note that the algebraic expressions in parentheses on both sides of the equality are not identical,

Therefore, we'll examine the difference between the expressions inside the parentheses, we want to bring the expression in parentheses on the right side to be identical to the expression in parentheses on the left side, let's note that between the two terms in parentheses on the left side and between the two terms in the expression in parentheses on the right side there is a clear proportion:

$\frac{18}{-9}=\frac{-2a}{a}=-2$ And additionally let's note that according to the multiplication rules:

$10=(-5)\cdot(-2)$Therefore, we can first rearrange (using the commutative property of addition) the expression in parentheses on the left side, in parallel we'll present the number 10 as a multiplication of numbers as mentioned above:

$10(a-9)\\ \downarrow\\ (-5)\cdot(-2)(-9+a)$Let's emphasize again that the sign preceding each term is an inseparable part of it, and therefore when using the commutative property (in addition) we switched the places of the terms along with their preceding signs.

Let's continue then and note that if we "insert" the factor $-2$into the parentheses, we'll get that the expressions in parentheses on both sides of the equality will be identical , we'll do this of course by applying the multiplication operation on the expression in parentheses, using the distributive property:

$x(y+z)=xy+xz$Let's apply this in the expression we got in the last stage, which is the expression on the right side in the given equation:

$(-5)\cdot(-2)(-9+a)\\ \downarrow\\ (-5)\big((-2)\cdot(-9)+(-2)\cdot a\big)\\ (-5)(18-2a)$In the last stage, we simplified the expression in parentheses,

Now let's return again to the original problem and summarize the development stages, we got that:

$?(18-2a)=10a-90 \\ \downarrow ?(18-2a)=10(a-9) \\ ?(18-2a)=(-5)(-2)(a-9) \\ ?(18-2a)=(-5)(18-2a) \\$

We got that the expressions in parentheses in the expressions on both sides of the equation are identical, therefore now we can easily complete the missing part (the factor marked on the left side with a question mark) and determine unequivocally that the correct answer is answer C.

Important note:

This problem is not an equation-solving problem, but a problem of completing the missing part, meaning - the solver is not required to find the value of the unknown (or unknowns) which when substituted in the equation will yield a true statement, but to find the missing algebraic expressions in the marked places and this in order to get equality between the algebraic expressions on both sides of the equation (i.e., regardless of the value of the unknown), therefore in the above problem (for example) although clearly for:

$a=9$indeed there is equality between the sides of the equation,

This information is of no use in order to answer what we were asked in the problem.

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:

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

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

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

To solve the problem, follow these steps:

• Step 1: Recognize the expression $21xy + 9$ needs to be matched by factoring it as a common product expression that includes $3y$.
• Step 2: Identify the greatest common factor in $21xy + 9$, which is $3$. Thus, the factorization is $3(7xy + 3)$.
• Step 3: Now, we need $3y(?-?) = 3(7xy + 3)$. Since we factor out a $3$, the matching terms should sum up to $y(7x) + y\left(\frac{-3}{y}\right)$.
• Step 4: Match the missing numbers found in the expression: $? - ? = 7x, \frac{-3}{y}$.

By matching, the factors yield $3y(?-?) = 3y(7x - \frac{3}{y})$. This confirms the missing values are $7x$ and $\frac{-3}{y}$.

Therefore, the correct completion of the expression is $7x, \frac{-3}{y}$, which corresponds to choice 4.

To solve this problem, we need to factor both sides of the equation $7y(?-?) = -14xy + 21$.

First, observe the terms on the right-hand side: $-14xy$ and $21$.

• Both terms share a common factor of 7. Factoring out 7 from both terms yields:

$7(-2xy + 3)$.

Thus, the expression becomes:

$7y(-2x + \frac{3}{y})$ since the y was outside the parenthesis in the left.

Matching terms with $7y(? - ?)$, the missing values are $\frac{3}{y}$ and $2x$.

Therefore, the solution is: $\frac{3}{y}, 2x$, corresponding to the correct choice.

To solve the problem, follow these steps:

Step 1: Start by analyzing the equation $?(5b - 3) = 20b - 15$.

Step 2: Note that the right side of the equation is already in the form of a subtraction: $20b - 15$.

Step 3: Factor the right side of the equation:

• Both terms on the right side, $20b$ and $-15$, can be expressed as multiples of 5. Therefore, factor out 5:

$20b - 15 = 5(4b - 3)$

Step 4: Compare with the left side of the equation:

$?(5b - 3)$

Step 5: While both sides aim to express multiplication that mirrors each other structurally, endeavoring to achieve a strict variable ratio equivalence through a diverse scale factor remains void due to differing algebraic expression. Therefore, direct factor parity isn't adhered.

Step 6: Since the expression inside the parentheses has a mismatch that prevents it from matching, algebraically, certain middle scale accessing (like $?$) will not yield a uniform factor.

Conclusion: Thus, if followed step-by-step identity accuracy and alignment of circumstances forego a consensus number-winning, thus deeming "No adequate solution" viable.

Therefore, the solution to the problem is No adequate solution.

To solve this problem, we'll rewrite the expression $12ab(?+?)=24abc+36$, focusing on the right-hand side, $24abc+36$.

Step 1: Factor the right-hand side:

Both terms on the right-hand side, $24abc$ and $36$, have a common factor. The greatest common factor (GCF) of $24abc$ and $36$ is $12$. Therefore, we can factor out $12$:

$24abc + 36 = 12(2ac + 3)$.

Step 2: Match the factored form with the left-hand side expression:

The equation now resembles $12ab(?+?) = 12(2ac + 3)$. To make the left-hand side equivalent to this expression, we equate it to the factorization result:

$12ab(?+?) = 12 \times (2ac + 3)$ implies $ab(?+?) = 2ac + 3$.

Step 3: Divide both sides by $ab$:

$? + ? = \frac{2ac}{ab} + \frac{3}{ab} = 2c + \frac{3}{ab}$.

Therefore, the missing values in the expression are $2c$ and $\frac{3}{ab}$.

Comparing this with the answer choices, the correct choice that aligns with these values is: $2c, \frac{3}{ab}$.

Therefore, the solution to the problem is $2c, \frac{3}{ab}$.

To solve this problem, we'll follow these steps:

• Step 1: Look at the right side of the equation, $6mn + 9m$, and factor out the common term $3m$.
• Step 2: Upon factoring, rewrite the expression as $3m(2n + 3)$.
• Step 3: Compare this factored form to the left side $3m(?+?)$.
• Step 4: Identify the terms inside the parentheses: $2n$ for the first place and $3$ for the second.

Following these steps:
Step 1: Recognize that both terms $6mn$ and $9m$ on the right have a common factor of $3m$.
Step 2: Factor out $3m$ to get $3m(2n + 3)$.
Step 3: Note that $3m(?+?)$ needs to match $3m(2n + 3)$.
Step 4: Thus, the expressions inside the parentheses are $? = 2n$ and $? = 3$.

Therefore, the solution to the problem is $2n, 3$.

To solve the problem of finding the missing values in the equation $(?+5)(3a+?)=6ax+15a-16x-40$, we need to expand the left side and equate the resulting expression with the right side.

Step-by-step solution:

• Step 1: Start by expanding $(?+5)(3a+?)$. Assume the missing values are $x$ and $b$ respectively, thus forming $(x+5)(3a+b)$.
• Step 2: Expand the expression on the left side:
  $(x+5)(3a+b) = x \cdot 3a + x \cdot b + 5 \cdot 3a + 5 \cdot b = 3ax + bx + 15a + 5b$.
• Step 3: Equate the expanded expression with the right side $6ax + 15a - 16x - 40$.

Upon comparing coefficients and constant terms:
- Coefficient of $ax$ should match: $3 = 6$. By substituting, $b = 2x$ to match terms.
- Constant term: $5b = -40$, therefore, solve for $b$. We find $b = -8$ because $5(-8) = -40$.

Once substitutions are made, verify that terms align. This shows:

• $x = -8$ (balancing constant $-40$) and substituting $b = 2x$ yields a consistent algebraic identity.

Therefore, the values that satisfy the equation are $(x, b) = (-8, 2x)$, confirming the answer $(-8, 2x)$. This choice best aligns when comparing the choices provided.