SAT-32: Factoring Advanced Quadratics (ax² + bx + c)

SAT-32: Factoring Advanced Quadratics (ax² + bx + c)

Description: In SAT-31 you factored quadratics where the leading number was 1. Now the front coefficient a is not 1 (for example 2x² + 7x + 3). This lesson teaches the AC method — a step-by-step trick that always works.

The AC Method (split the middle)

(Oʻzbekcha: AC usuli — bosh koeffitsiyent 1 ga teng boʻlmagan kvadrat ifodalarni koʻpaytuvchilarga ajratishning ishonchli yoʻli.)

For a quadratic ax² + bx + c:

1. Multiply a × c.
2. Find two numbers that multiply to a·c and add to b.
3. Split the middle term bx into those two numbers.
4. Factor by grouping (see SAT-29).

Rule: a·c gives you what to multiply to, and b gives you what to add to. (Oʻzbekcha: ikkita son a·c ga koʻpaytirilib, b ga qoʻshilishi kerak.)

Worked Example

Factor 2x² + 7x + 3.

• a·c = 2 × 3 = 6. We need two numbers that multiply to 6 and add to 7 → 6 and 1.
• Split: 2x² + 6x + 1x + 3
• Group: 2x(x + 3) + 1(x + 3)
• Factor out (x + 3): (2x + 1)(x + 3)

Check by FOIL: (2x + 1)(x + 3) = 2x² + 7x + 3. ✓

(Oʻzbekcha: oʻrta hadni (7x) ikkita hadga ajratamiz, soʻng guruhlab umumiy koʻpaytuvchini chiqaramiz.)

Watch the signs

If c is negative, the two numbers have opposite signs. If b is negative but c is positive, both numbers are negative.

(Oʻzbekcha: agar c manfiy boʻlsa, sonlar har xil ishorada boʻladi; ishoralarga juda eʼtibor bering.)

Practice

Factor 3x² − 10x + 8.

Key words — Kalit soʻzlar

• Factor — koʻpaytuvchilarga ajratish
• Quadratic — kvadrat ifoda / tenglama
• Coefficient — koeffitsiyent
• Leading coefficient — bosh koeffitsiyent
• Term — had
• Middle term — oʻrta had
• Grouping — guruhlash
• Product — koʻpaytma
• Sum — yigʻindi
• Sign — ishora

Summary

• AC method: find two numbers that multiply to a·c and add to b.
• Split the middle term, then factor by grouping.
• Always check by multiplying back (FOIL).
• Signs: negative c → opposite signs; positive c with negative b → both negative.