Solve (x-?)(x+?) = x²+x-12: Find the Missing Numbers in Factored Form

Q: How do I find two numbers that multiply to -12 and add to 1?

List all factor pairs of -12: (1,-12), (-1,12), (2,-6), (-2,6), (3,-4), (-3,4). Then check which pair adds to 1. Only -3 and 4 work: -3 + 4 = 1!

Q: Why is the answer (x-3)(x+4) and not (x+3)(x-4)?

The key is the middle term coefficient! When you expand (x+3)(x-4), you get x² - x - 12, but we need x² + x - 12. The signs matter!

Q: How do I remember which number goes with which sign?

Think about the expanded form: $ (x-a)(x+b) = x^2 + (b-a)x - ab $. The middle term is (b-a), so if you need +1, then b-a = 1, meaning b = a+1.

Q: Can I check my answer without expanding?

Yes! Substitute a simple value like x = 0 into both sides. For our answer: (0-3)(0+4) = -12, and 0² + 0 - 12 = -12. They match! ✓

Quadratic Factoring with Missing Terms

$(x-?)(x+?)=x^2+x-12$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the appropriate numbers

00:03 Find the trinomial coefficients

00:14 We want to find 2 numbers whose product equals coefficient C

00:22 and their sum equals coefficient B

00:27 These are the appropriate numbers, let's substitute in the trinomial

00:31 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

$(x-?)(x+?)=x^2+x-12$

Step-by-step solution

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

Step 1: Determine two numbers whose product is $-12$ (the constant term) and whose sum is $1$ (the coefficient of $x$ ).
Step 2: Verify these numbers are $-3$ and $4$ . Since $(-3) \times 4 = -12$ and $(-3) + 4 = 1$ , both conditions are met.
Step 3: Substitute these numbers into the expression $(x-?)(x+?)$ to get $(x-3)(x+4)$ .
Step 4: Double-check the factorization by expanding $(x-3)(x+4)$ to ensure it results in $x^2 + x - 12$ .

Therefore, the factorized form of the given quadratic polynomial is $(x-3)(x+4)$ .

Final Answer

$\left(x-3\right)\left(x+4\right)$

Key Points to Remember

Essential concepts to master this topic

Factor Pairs: Find two numbers whose product is -12 and sum is 1
Technique: -3 and 4 work because (-3) × 4 = -12 and -3 + 4 = 1
Check: Expand (x-3)(x+4) to verify it equals x² + x - 12 ✓

Common Mistakes

Avoid these frequent errors

Confusing the signs in the factored form
Don't assume the signs match the factor pair directly = wrong factorization! The signs in (x-a)(x+b) depend on which number is subtracted and which is added. Always check that your factored form expands correctly to match the original polynomial.

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 find two numbers that multiply to -12 and add to 1?

List all factor pairs of -12: (1,-12), (-1,12), (2,-6), (-2,6), (3,-4), (-3,4). Then check which pair adds to 1. Only -3 and 4 work: -3 + 4 = 1!

Why is the answer (x-3)(x+4) and not (x+3)(x-4)?

The key is the middle term coefficient! When you expand (x+3)(x-4), you get x² - x - 12, but we need x² + x - 12. The signs matter!

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

Double-check your factor pairs of the constant term. If none work, the quadratic might not factor nicely with integers, or you may need to use the quadratic formula instead.

How do I remember which number goes with which sign?

Think about the expanded form: $(x-a)(x+b) = x^2 + (b-a)x - ab$ . The middle term is (b-a), so if you need +1, then b-a = 1, meaning b = a+1.

Can I check my answer without expanding?

Yes! Substitute a simple value like x = 0 into both sides. For our answer: (0-3)(0+4) = -12, and 0² + 0 - 12 = -12. They match! ✓

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