SAT-Math-31: Factoring Standard Quadratics ($x^2 + bx + c$)

1. The Product-Sum Framework

When you have a quadratic expression in standard form with a leading coefficient of 1:

$$x^2 + bx + c$$

Factoring it means rewriting it into two binomials:

$$(x + r)(x + s)$$

If you expand $(x + r)(x + s)$ using FOIL, you get $x^2 + (r+s)x + rs$. By matching this back to our standard form, we get our ultimate factoring mission:

1. Find two numbers that multiply to the constant term ($c$).

2. Those same two numbers must add up to the middle coefficient ($b$).

Sign Distribution Cheat Sheet

• If $c$ is positive, your two numbers will share the same sign (either both positive or both negative, matching the sign of $b$).

• If $c$ is negative, your two numbers will have opposite signs (the larger number takes the sign of $b$).

🇺🇿 Uzbek Explanation:

Kvadrat uchhadni ko'paytuvchilarga ajratish ($x^2 + bx + c$): > Oldida son bo'lmagan ($1$ bo'lgan) uchhadni ko'paytuvchilarga ajratish uchun biz ikkita shunday son topishimiz kerakki:

1. Ularni ko'paytirganda oxiridagi son ($c$) chiqishi kerak.

2. Ularni qo'shganda esa o'rtadagi son ($b$) kelib chiqishi kerak.

   Masalan, $x^2 + 5x + 6$ ifodada ko'paytmasi $6$, yig'indisi $5$ bo'lgan sonlar $2$ va $3$ dir. Javob: $(x + 2)(x + 3)$.

2. High-Yield SAT Practice Question

Question: Which of the following is an equivalent form of the quadratic expression $x^2 - 4x - 12$?

A) $(x - 6)(x + 2)$

B) $(x + 6)(x - 2)$

C) $(x - 4)(x + 3)$

D) $(x - 12)(x + 1)$

Step-by-Step Explanation:

1. Identify your target targets:

   • Target Product ($c$) = $-12$

   • Target Sum ($b$) = $-4$

2. Analyze the signs: Since the product is negative ($-12$), our two magic numbers must have opposite signs (one positive, one negative).

3. List factor pairs of 12 and test their differences:

   • $1$ and $12 \rightarrow$ difference is $11$ (No)

   • $3$ and $4 \rightarrow$ difference is $1$ (No)

   • $6$ and $2 \rightarrow$ difference is $4$ (Bingo!)

4. Assign the signs carefully: The sum must be $-4$. This means the larger number must be negative:

   • $-6 \cdot 2 = -12$

   • $-6 + 2 = -4$

5. Write the binomial factors: $(x - 6)(x + 2)$.

Correct Answer: A

Summary

• The Core Rule: Find two factors of the constant $c$ that add up to the linear coefficient $b$.

• Let the sign of $c$ guide you: positive $c$ means same signs; negative $c$ means different signs.

• Always do a quick FOIL expansion in your head to check your signs before selecting your final answer.

Outstanding work as always, my bro! Let's lock in the next level. Ready for SAT-Math-32: Factoring Complex Quadratics ($ax^2 + bx + c$ via Slip & Slide / AC Method)?