Factorize the Quadratic x² - 3x - 4: Understanding Product Representation

Q: How do I know which two numbers multiply to -4 and add to -3?

List all factor pairs of -4: (1, -4), (-1, 4), (2, -2). Then check which pair adds to -3. Only -4 + 1 = -3, so the numbers are -4 and 1.

Q: Why do I get the opposite signs in the factored form?

When your factors are -4 and 1, you write (x - 4)(x + 1). The signs are opposite because if x - 4 = 0, then x = 4, and if x + 1 = 0, then x = -1.

Q: Can I use the quadratic formula instead of factoring?

Yes! The quadratic formula always works, but factoring is faster when the quadratic factors nicely. Use factoring first, then try the formula if factoring doesn't work easily.

Q: How do I check if my factored form is correct?

Expand your answer using FOIL: $(x-4)(x+1) = x^2 + x - 4x - 4 = x^2 - 3x - 4$. If this matches the original, you're right!

Q: What if the coefficient of x² isn't 1?

When a ≠ 1, you still find two numbers that multiply to ac and add to b. The grouping method works the same way, just with more steps.

Quadratic Factoring with Grouping Method

Find the representation of the product of the following function

$f(x)=x^2-3x-4$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Convert trinom to multiplication

00:03 Identify the trinom coefficients

00:06 We want to find 2 numbers whose sum equals coefficient B

00:11 and their product equals coefficient C

00:17 These are the matching numbers, let's substitute in multiplication

00:26 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

Find the representation of the product of the following function

$f(x)=x^2-3x-4$

Step-by-step solution

To solve this problem, let's follow these steps:

Step 1: Identify the coefficients: $a = 1$ , $b = -3$ , $c = -4$ .
Step 2: Calculate the product $ac = 1 \times -4 = -4$ and the sum $b = -3$ .
Step 3: Find two numbers that multiply to $-4$ and add to $-3$ . These numbers are $-4$ and $1$ .
Step 4: Rewrite the middle term $-3x$ using $-4$ and $1$ as $-4x + x$ .
Step 5: Factor by grouping: $(x^2 - 4x) + (x - 4)$ .
Step 6: Factor out common factors: $x(x - 4) + 1(x - 4)$ .
Step 7: Notice the common binomial factor: $(x - 4)(x + 1)$ .

Therefore, the factored form of the quadratic function $f(x) = x^2 - 3x - 4$ is $(x - 4)(x + 1)$ , which corresponds to choice 3.

Final Answer

$(x-4)(x+1)$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find two numbers that multiply to ac and add to b
Grouping Technique: Rewrite $-3x$ as $-4x + x$ using factors -4 and 1
Verification: Expand $(x-4)(x+1)$ to get $x^2-3x-4$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong signs in factored form
Don't write (x+4)(x-1) when you found factors -4 and +1 = wrong factorization! This gives x²-3x-4 when expanded, not the original. Always check: if your factors are -4 and +1, write (x-4)(x+1) with opposite signs.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=2,b=2,c=2 $$

FAQ

Everything you need to know about this question

How do I know which two numbers multiply to -4 and add to -3?

List all factor pairs of -4: (1, -4), (-1, 4), (2, -2). Then check which pair adds to -3. Only -4 + 1 = -3, so the numbers are -4 and 1.

Why do I get the opposite signs in the factored form?

When your factors are -4 and 1, you write (x - 4)(x + 1). The signs are opposite because if x - 4 = 0, then x = 4, and if x + 1 = 0, then x = -1.

Can I use the quadratic formula instead of factoring?

Yes! The quadratic formula always works, but factoring is faster when the quadratic factors nicely. Use factoring first, then try the formula if factoring doesn't work easily.

How do I check if my factored form is correct?

Expand your answer using FOIL: $(x-4)(x+1) = x^2 + x - 4x - 4 = x^2 - 3x - 4$ . If this matches the original, you're right!

What if the coefficient of x² isn't 1?

When a ≠ 1, you still find two numbers that multiply to ac and add to b. The grouping method works the same way, just with more steps.

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Factorize the Quadratic x² - 3x - 4: Understanding Product Representation

Quadratic Factoring with Grouping Method

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which two numbers multiply to -4 and add to -3?

Why do I get the opposite signs in the factored form?

Can I use the quadratic formula instead of factoring?

How do I check if my factored form is correct?

What if the coefficient of x² isn't 1?

🌟 Unlock Your Math Potential

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