Exploring Algebraic Identities | Exercise 4.4

Question 1

Fill in the blanks to complete the following identities:

Question diagram 1
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Solution
Understand the Question
  • To factorize a quadratic expression of the form ax2+bx+cax^2 + bx + c by splitting the middle term:
    1. Find two numbers whose product is a×ca \times c and whose sum is bb.
    2. Split the middle term bxbx using these two numbers.
    3. Factor by grouping common terms.

(i) s2−11s+24s^2 - 11s + 24

Step 1 · Factorize by Splitting the Middle Term

Diagram 1

Find two numbers whose product is 2424 and sum is −11-11. The numbers are −3-3 and −8-8.

s2−11s+24=s2−3s−8s+24=s(s−3)−8(s−3)=(s−3)(s−8)\begin{aligned} s^2 - 11s + 24 &= s^2 - 3s - 8s + 24 \\ &= s(s - 3) - 8(s - 3) \\ &= (s - 3)(s - 8) \end{aligned}
Answer

(i) (s−3)(s−8)(s - 3)(s - 8)

(ii) 3x2−4x−73x^2 - 4x - 7

Step 1 · Factorize by Splitting the Middle Term

Diagram 2

Find two numbers whose product is 3×(−7)=−213 \times (-7) = -21 and sum is −4-4. The numbers are −7-7 and 33.

3x2−4x−7=3x2−7x+3x−7=x(3x−7)+1(3x−7)=(3x−7)(x+1)\begin{aligned} 3x^2 - 4x - 7 &= 3x^2 - 7x + 3x - 7 \\ &= x(3x - 7) + 1(3x - 7) \\ &= (3x - 7)(x + 1) \end{aligned}
Answer

(ii) (3x−7)(x+1)(3x - 7)(x + 1)

(iii) 10x2−11x−610x^2 - 11x - 6

Step 1 · Factorize by Splitting the Middle Term

Diagram 3

Find two numbers whose product is 10×(−6)=−6010 \times (-6) = -60 and sum is −11-11. The numbers are −15-15 and 44.

10x2−11x−6=10x2−15x+4x−6=5x(2x−3)+2(2x−3)=(2x−3)(5x+2)\begin{aligned} 10x^2 - 11x - 6 &= 10x^2 - 15x + 4x - 6 \\ &= 5x(2x - 3) + 2(2x - 3) \\ &= (2x - 3)(5x + 2) \end{aligned}
Answer

(iii) (2x−3)(5x+2)(2x - 3)(5x + 2)

(iv) 6x2+7x+26x^2 + 7x + 2

Step 1 · Factorize by Splitting the Middle Term

Diagram 4

Find two numbers whose product is 6×2=126 \times 2 = 12 and sum is 77. The numbers are 33 and 44.

6x2+7x+2=6x2+3x+4x+2=3x(2x+1)+2(2x+1)=(3x+2)(2x+1)\begin{aligned} 6x^2 + 7x + 2 &= 6x^2 + 3x + 4x + 2 \\ &= 3x(2x + 1) + 2(2x + 1) \\ &= (3x + 2)(2x + 1) \end{aligned}
Answer

(iv) (3x+2)(2x+1)(3x + 2)(2x + 1)

Common Mistakes
  • Sign Errors in Factors: Ensure the two chosen numbers satisfy both conditions: product =ac= ac and sum =b= b, taking careful note of negative signs.
  • Grouping Sign Errors: When factoring out a negative sign (e.g., −8(s−3)-8(s - 3)), ensure the sign inside the bracket flips correctly.

More questions in Exercise 4.4

Q1

Fill in the blanks to complete the following identities:

Q2

Select and use the identity that will help you find the following products without multiplying directly:

(i) (41)2(41)^2

(ii) (27)2(27)^2

(iii) (23×17)(23 \times 17)

(iv) (135)2(135)^2

(v) (97)2(97)^2

(vi) (18×29)(18 \times 29)

(vii) (34×43)(34 \times 43)

(viii) (205)2(205)^2

Q3

Factor the following:

(i) 9a2+b2+4c2−6ab+12ac−4bc9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc

(ii) 16s2+25t2−40st16s^2 + 25t^2 - 40st

(iii) r2−r−42r^2 - r - 42

(iv) 49g2+14gh+h249g^2 + 14gh + h^2

(v) 64u2+121v2+4w2−176uv−32uw+44vw64u^2 + 121v^2 + 4w^2 - 176uv - 32uw + 44vw

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