Solve (a+?)(−2a+4) = −2a²−20a+48: Find the Missing Term

Q: How do I know which terms to match together?

Match terms with the same degree! Compare $ a^2 $ terms together, $ a $ terms together, and constant terms together. This ensures you're solving the right equations.

Q: What if I get different answers from different terms?

If matching different terms gives different values for the missing element, then no solution exists. The equation cannot be satisfied with any single value.

Q: Can I work backwards from the expanded form?

Yes! You can factor the right side first: $ -2a^2 - 20a + 48 = -2(a^2 + 10a - 24) = -2(a + 12)(a - 2) $. Then compare with the given form.

Q: Why do we expand instead of factoring directly?

Expanding is often more reliable because it gives you a systematic way to match coefficients. Factoring can be tricky and you might miss solutions or make algebraic errors.

Q: What if the coefficient of the quadratic term doesn't match?

Check your expansion! The $ a^2 $ terms should match automatically if you expanded correctly. If they don't, you made an error in the distributive property step.

Polynomial Factoring with Missing Coefficients

Complete the missing element

$(a+?)(-2a+4)=-2a^2-20a+48$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing term

00:03 Let X be the unknown

00:12 Open parentheses properly, multiply each factor by each factor

00:27 Simplify what's possible, and collect like terms

00:55 Factor out the common term

01:04 Isolate the unknown X

01:18 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

Complete the missing element

$(a+?)(-2a+4)=-2a^2-20a+48$

Step-by-step solution

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

Step 1: Expand the expression $(a + ?)(-2a + 4)$ using the distributive property.
Step 2: Equate the resulting expression with the given form $-2a^2 - 20a + 48$ .
Step 3: Solve for the missing '?' that satisfies the equation.

Step 1: Expand the original expression:

$(a + ?)(-2a + 4) = a(-2a) + a(4) + ?(-2a) + ?(4)$

Expanding gives us:

$-2a^2 + 4a - 2a? + 4?$

Step 2: Equate this with the provided expanded form $-2a^2 - 20a + 48$ :

$-2a^2 + 4a - 2a? + 4? = -2a^2 - 20a + 48$

Step 3: Match the coefficients:

Compare the linear terms $4a - 2a?$ with $-20a$ :
$4 - 2? = -20$ implies $4 = -20 + 2?$
Solve for $?$ :
$2? = 24$ gives $? = 12$

The missing element is $12$ .

Final Answer

$12$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Use distributive property to multiply each term systematically
Coefficient Matching: Compare like terms: 4a - 2a(12) = -20a
Verification: Substitute ? = 12 back into original expression ✓

Common Mistakes

Avoid these frequent errors

Not matching coefficients systematically
Don't just guess values for the missing term = wrong answers every time! Students often try random numbers without expanding properly. Always expand the left side completely, then match each coefficient (constant, linear, quadratic) with the right side.

Practice Quiz

Test your knowledge with interactive questions

$(3+20)\times(12+4)=$

FAQ

Everything you need to know about this question

How do I know which terms to match together?

Match terms with the same degree! Compare $a^2$ terms together, $a$ terms together, and constant terms together. This ensures you're solving the right equations.

What if I get different answers from different terms?

If matching different terms gives different values for the missing element, then no solution exists. The equation cannot be satisfied with any single value.

Can I work backwards from the expanded form?

Yes! You can factor the right side first: $-2a^2 - 20a + 48 = -2(a^2 + 10a - 24) = -2(a + 12)(a - 2)$ . Then compare with the given form.

Why do we expand instead of factoring directly?

Expanding is often more reliable because it gives you a systematic way to match coefficients. Factoring can be tricky and you might miss solutions or make algebraic errors.

What if the coefficient of the quadratic term doesn't match?

Check your expansion! The $a^2$ terms should match automatically if you expanded correctly. If they don't, you made an error in the distributive property step.

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