Factoring the Quadratic: Find the Product Representation of f(x) = x² - 3x - 18

Q: Why do we need two numbers that multiply to -18 AND add to -3?

This comes from the factoring method! When you expand $ (x + m)(x + n) $, you get $ x^2 + (m+n)x + mn $. So m + n must equal the middle coefficient and mn must equal the constant term.

Q: How do I handle the negative signs when factoring?

Look at the signs carefully! Since the constant is negative (-18), one factor must be positive and one negative. The larger absolute value takes the sign that matches the middle term's sign.

Q: What if I can't find factor pairs that work?

List all factor pairs of the constant systematically: ±1×±18, ±2×±9, ±3×±6. Check each pair's sum. If none work, the quadratic might not factor with integers or you may have made an error.

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

Yes! The quadratic formula works for any quadratic, but factoring is often faster when the numbers work out nicely. Plus, factored form makes it easy to find the zeros: x = 6 and x = -3.

Q: How do I know which factor goes with which sign?

For $ x^2 - 3x - 18 $, you need factors of -18 that add to -3. Since 3 + (-6) = -3, you get $ (x - 6)(x + 3) $. The signs follow from the sum requirement.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's convert this trinomial into a multiplication problem.

00:12 First, identify the coefficients from the trinomial.

00:16 Next, find two numbers whose sum is equal to coefficient B.

00:21 Also, make sure their product equals coefficient C.

00:28 These are the numbers we need. We'll use them in our multiplication.

00:38 And that's how we find the solution to the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Find the representation of the product of the following function

$f(x)=x^2-3x-18$

2

Step-by-step solution

To solve the problem of factoring the quadratic expression $f(x) = x^2 - 3x - 18$ , we will use the following method:

Step 1: Identify and understand the quadratic expression, which is given in standard form: $ax^2 + bx + c$ . For this expression, $a = 1$ , $b = -3$ , and $c = -18$ .
Step 2: Compute the product of $a$ and $c$ , which yields $1 \cdot (-18) = -18$ . We need to find two numbers whose product is $-18$ and whose sum is $-3$ .
Step 3: Look for pairs of factors of $-18$ : - $1, -18$ - $-1, 18$ - $2, -9$ - $-2, 9$ - $3, -6$ - $-3, 6$

Among these, the pair $(3, -6)$ adds up to $-3$ and multiplies to $-18$ .

Step 4: Rewrite the quadratic expression using these numbers to represent the middle term:
$x^2 - 3x - 18 = x^2 + 3x - 6x - 18$ .
Step 5: Group the terms to facilitate factoring:
$(x^2 + 3x) + (-6x - 18)$ .
Step 6: Factor out the common factors in each grouped terms:
$x(x + 3) - 6(x + 3)$ .
Step 7: Factor out the common binomial:
$(x - 6)(x + 3)$ .

Therefore, the factorized form of the quadratic function $f(x) = x^2 - 3x - 18$ is $(x - 6)(x + 3)$ .

3

Final Answer

$(x-6)(x+3)$

Key Points to Remember

Essential concepts to master this topic

Factor pairs: Find two numbers that multiply to -18 and add to -3
Technique: Use 3 and -6 since 3 × (-6) = -18 and 3 + (-6) = -3
Check: Expand (x - 6)(x + 3) = x² - 3x - 18 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check both sum and product conditions
Don't just find factors of -18 without checking their sum = wrong factorization! For example, -1 and 18 multiply to -18 but add to 17, not -3. Always verify both conditions: product equals c AND sum equals b.

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

Why do we need two numbers that multiply to -18 AND add to -3?

+

This comes from the factoring method! When you expand $(x + m)(x + n)$ , you get $x^2 + (m+n)x + mn$ . So m + n must equal the middle coefficient and mn must equal the constant term.

How do I handle the negative signs when factoring?

+

Look at the signs carefully! Since the constant is negative (-18), one factor must be positive and one negative. The larger absolute value takes the sign that matches the middle term's sign.

What if I can't find factor pairs that work?

+

List all factor pairs of the constant systematically: ±1×±18, ±2×±9, ±3×±6. Check each pair's sum. If none work, the quadratic might not factor with integers or you may have made an error.

Can I use the quadratic formula instead of factoring?

+

Yes! The quadratic formula works for any quadratic, but factoring is often faster when the numbers work out nicely. Plus, factored form makes it easy to find the zeros: x = 6 and x = -3.

How do I know which factor goes with which sign?

+

For $x^2 - 3x - 18$ , you need factors of -18 that add to -3. Since 3 + (-6) = -3, you get $(x - 6)(x + 3)$ . The signs follow from the sum requirement.

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Factoring the Quadratic: Find the Product Representation of f(x) = x² - 3x - 18

Quadratic Factoring with Negative Constants

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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

Why do we need two numbers that multiply to -18 AND add to -3?

How do I handle the negative signs when factoring?

What if I can't find factor pairs that work?

Can I use the quadratic formula instead of factoring?

How do I know which factor goes with which sign?

🌟 Unlock Your Math Potential

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