Also, ACHF ABGF - CBGH = (ABDE – FGDE) – (CBDK – HGDK) = = ABDE - FGDE - CBDK+HGDK. Hence, (a - b)2=a2 ab ab+b2 a2 - 2 ab+b2. = NOTE. The steps in the geometric proof exactly correspond with the steps in the algebraic process of multiplying a - b by a b. This proof illustrates the truth of the Law of Signs in multiplication. 89. Multiply a + b by a−b. a + b The required product is the difference between a(a+b) and b(a+b). Hence, The product of the sum and difference of two numbers is equal to the difference of the squares of the numbers. Geometric Proof. Let AB = a, BC= b. (III) Upon AB describe the square ABKE, Fig. 3. Make EG=b. Draw GH, CD at right angles to AE and AC respectively; prolong EK. Also, ACHGACDE- GHDE. =(ABKE+BCDK)–(GLKE+LHDK) = ABKE+ BCDK - GLKE-LHDK. Hence, (a+b) ( a − b) = a2 + ab — ab — b2 — a2 — b2. = Example 1. Expand by statement (I) 832. SOLUTION. 832 = (80+3)2 = 802 + 2(80 × 3) + 32 Example 2. Expand by statement (II) 672. 672 = (70 — 3)2 = 702 — 2(3 × 70) + 32 = 4900 420 + 9 = 4489. Example 3. Multiply 43 by 37. x = 43 × 37 (40+3)(403)=402-321600-9=1591. For other important identities, see page 143, examples 42 to 50 inclusive. 52. (3x2+7x) (3 x2 — 7 x). Since (a + b)2 = a2 + 2 ab + b2, hence, any trinomial expression consisting of the sum of the squares of two quantities plus twice their product can be resolved into two equal factors, each of which is the sum of the two quantities. Example 1. Factor 9x2+30 xy + 25 y2. 9 x2 = (3 x)2. 25 y2 = (5 y)2. 30 xy=2(3x × 5 y). Therefore, 9 x2 + 30 xy + 25 y2 = (3 x + 5 y)2. Factor: 1. x2+2x+1. 2. x2+4x+4. 3. x2+10x + 25. 4. a2+14a+49. 5. a2+20 a + 100. EXERCISE 74 6. 4x2+4 xy + y2. 7. 9x2+6xy+ y2. 8. 16 x2+24 xy + 9 y2. 9. 25 x2+20 xy + 4 y2. 10. 25 x2+40 xy + 16 y2. 11. 49 a2+14 ab + b2. 91. The square of a difference. Since (a - b)2= a2 — 2 ab + b2, a trinomial consisting of the sum of the squares of two quantities diminished by twice their product can be resolved into two factors, each of which is the difference of the two quantities. Example 1. Factor 16x4 - 24x2y2 + 9 y1. Therefore, 164 — 24 x2y2 + 9 y2 = (4 x2 — 3 y2)2. Factor: 1. 4x2-4x+1. 2. 9x2-6x+1. EXERCISE 75 3. x2-2xy + y2. 4. 16x2-8 xy + y2. |