Decompose the Expression 5x^2+9x+4 into Trinomials

Q: Why do I multiply a and c first in the AC method?

The product ac = 20 helps you find the right factor pair. You need two numbers that multiply to 20 AND add to the middle coefficient 9. This systematic approach prevents guessing!

Q: How do I know which way to group the terms?

After splitting the middle term into $ 5x^2 + 5x + 4x + 4 $, group consecutive terms: (5x² + 5x) + (4x + 4). Look for common factors in each group.

Q: Can I factor this expression differently?

No! When factoring over integers, there's only one correct factorization (ignoring order). $ (5x+4)(x+1) $ is the unique answer for this quadratic.

Q: How do I check if my factorization is correct?

Use FOIL to expand your answer: First + Outer + Inner + Last. If you get back the original expression $ 5x^2+9x+4 $, your factorization is correct!

Quadratic Factoring with AC Decomposition Method

Decompose the following expression into trinomials

$5x^2+9x+4$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Break down the trinomial

00:03 Take out the common factor from parentheses

00:17 Reduce what's possible

00:23 Identify the coefficients

00:33 Use the roots formula to find possible solutions

00:42 Substitute appropriate values and solve to find the solutions

00:57 Calculate the squares and products

01:14 Find the 2 possible solutions (addition and subtraction)

01:17 These are the solutions, with them we'll reconstruct the trinomial

01:21 Find what zeroes each solution

01:27 These will be the factors of the trinomial

01:46 Substitute this trinomial back into the equation

01:54 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Decompose the following expression into trinomials

$5x^2+9x+4$

Step-by-step solution

To solve the problem of factoring the quadratic expression $5x^2 + 9x + 4$ , we can use the decomposition method:

Step 1: Identify $a = 5$ , $b = 9$ , and $c = 4$ . Calculate the product $a \cdot c = 5 \cdot 4 = 20$ .
Step 2: We need two numbers whose product is 20 and whose sum is 9. The numbers 5 and 4 satisfy this condition because $5\times4=20$ and $5+4=9$ .
Step 3: Decompose the middle term using these numbers: $5x^2 + 5x + 4x + 4$ .
Step 4: Group the terms: $(5x^2 + 5x) + (4x + 4)$ .
Step 5: Factor each group: $5x(x + 1) + 4(x + 1)$ .
Step 6: Use the distributive property to factor out the common binomial: $(5x + 4)(x + 1)$ .

Therefore, the factorization of the expression $5x^2 + 9x + 4$ is $(5x+4)(x+1)$ .

Final Answer

$(5x+4)(x+1)$

Key Points to Remember

Essential concepts to master this topic

AC Method: Find two numbers that multiply to ac and add to b
Decomposition: Split 9x as 5x + 4x since 5 × 4 = 20 and 5 + 4 = 9
Verification: Expand $(5x+4)(x+1) = 5x^2+9x+4$ ✓

Common Mistakes

Avoid these frequent errors

Finding wrong factor pairs for ac product
Don't use factors like 10 and 2 that multiply to 20 but don't add to 9! This leads to incorrect decomposition and wrong factorization. Always verify both conditions: the two numbers must multiply to ac AND add to b coefficient.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why do I multiply a and c first in the AC method?

The product ac = 20 helps you find the right factor pair. You need two numbers that multiply to 20 AND add to the middle coefficient 9. This systematic approach prevents guessing!

What if I can't find two numbers that work?

If no integer pairs multiply to ac and add to b, the quadratic might not factor nicely over integers. Double-check your arithmetic first, then consider if the problem requires the quadratic formula instead.

How do I know which way to group the terms?

After splitting the middle term into $5x^2 + 5x + 4x + 4$ , group consecutive terms: (5x² + 5x) + (4x + 4). Look for common factors in each group.

Can I factor this expression differently?

No! When factoring over integers, there's only one correct factorization (ignoring order). $(5x+4)(x+1)$ is the unique answer for this quadratic.

How do I check if my factorization is correct?

Use FOIL to expand your answer: First + Outer + Inner + Last. If you get back the original expression $5x^2+9x+4$ , your factorization is correct!

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