Algebraic Tiles Factorization Tool
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Comprehensive Guide to Quadratic Factorization
Mastering Middle Term Splitting & Grouping (English & Nepali Methods)
Key Rules for Mid-Term Splitting (ax² + bx + c)
1. Calculate Product and Middle Term:
Arrange the expression in standard form ax² + bx + c. Find the product of the first coefficient (a) and constant term (c): Product = a × c.
2. Assign Middle Term Sign First:
Always write down the sign of the middle term (b) immediately after the ax² term before choosing signs for factors.
3. Place Greater Factor First:
Find two factors, p and q, whose product equals a × c. Always place the greater numerical factor first right after the middle term sign.
4. Determine Sign for Smaller Factor:
Select either plus (+) or minus (-) for the smaller factor such that their combination satisfies the exact middle coefficient (b).
5. Grouping and Common Binomial Factor:
Split into 4 terms, group the first two and last two terms, factor out the common GCD from each pair, and extract the matching binomial factor to complete factorization.
Product: 1 × (-8) = -8 | Middle Term: +2
Factors of 8: 4 and 2 (Greater factor 4 placed first)
Sign Check: +4x and -2x satisfies middle term +2x.
= x² + 4x - 2x - 8
= x(x + 4) - 2(x + 4)
= (x + 4)(x - 2)
Product: 3 × (-8) = -24 | Middle Term: -5
Factors of 24: 8 and 3 (Greater factor 8 placed first)
Sign Check: Middle sign (-), so -8x and +3x satisfies -5x.
= 3x² - 8x + 3x - 8
= x(3x - 8) + 1(3x - 8)
= (3x - 8)(x + 1)
Middle Term Splitting का सुवर्ण नियमहरू (ax² + bx + c)
१. गुणनफल र मध्य पद पहिचान गर्ने:
वर्ग समीकरणलाई ax² + bx + c को standard form मा राख्नुहोस्। पहिलो पदको गुणक (a) र अन्तिम अङ्क (c) को गुणनफल पत्ता लगाउनुहोस् (Product = a × c)।
२. Middle Term को Sign पहिला लेख्ने:
Split गर्दा सधैँ Middle term को चिन्ह (Sign) लाई ax² पदपछि जस्ताको तस्तै लेख्नुहोस्।
३. ठूलो Factor अगाडि राख्ने:
गुणनफल (a × c) बाट प्राप्त दुईवटा गुणनखण्डहरू (factors) मध्ये ठूलो अङ्कलाई सधैँ अगाडि (First) राख्नुहोस्।
४. सानो Factor को Sign छनोट गर्ने:
सानो अङ्कको अगाडि प्लस (+) वा माइनस (-) मध्ये कुन राख्दा Middle term संतुष्ट (satisfy) हुन्छ, सोही चिन्ह छनोट गर्नुहोस्।
५. साझा (Common) लिने र खण्डीकरण गर्ने:
४ वटा पद बनेपछि पहिलो दुई पद र अन्तिम दुई पदबाट साझा (Common) तान्नुहोस्। दुवैतर्फ साझा कोष्ठक (Binomial factor) आइसकेपछि त्यसलाई पुनः बाहिर निकालेर खण्डीकरण पूरा गर्नुहोस्।
Product: 1 × (-8) = -8 | Middle Term: +2
Factors of 8: 4 र 2 (ठूलो अङ्क 4 अगाडि)
Sign Check: +4x र -2x गर्दा +2x satisfy हुन्छ।
= x² + 4x - 2x - 8
= x(x + 4) - 2(x + 4)
= (x + 4)(x - 2)
Product: 3 × (-8) = -24 | Middle Term: -5
Factors of 24: 8 र 3 (ठूलो अङ्क 8 अगाडि)
Sign Check: Middle sign (-), त्यसैले -8x र +3x ले -5x दिन्छ।
= 3x² - 8x + 3x - 8
= x(3x - 8) + 1(3x - 8)
= (3x - 8)(x + 1)