Algebraic expressions Questions and Answers

Algebraic expressions Questions

1. Simplify  (3mks)

2. Five year ago, a mother’s age was four times that of her daughter. In four years to come,

 she will be 2 ½ times the age of her daughter. Calculate the sum of their present ages

3. Mutua bought 160 trays of 8 eggs each at shs.150 per tray. On transportation 12 eggs

broke. He later discovered that 20 eggs were rotten. If he sold the rest at shs.180 per

tray, how much profit did he make?

4. Simplify;

 (a) 6a – 2b + 7a – 4b + 2

 (b) 2x – 2 3x + 2

2x 4x

5. Simplify 6x²y² + 13xy-5

 3x²y² – 13xy + 4

6. Given that x + y = 8 and x² + y² = 24

 Find;

 (a) the value of x² + 2xy + y²

 (b) Find the value of 2xy

 (c) x² – 2xy + y²

 (d) x – y

 (e) Value of x and y

7. Simplify the expression.

6x² + 35x – 6

2x² – 72

8. Simplify the expression

²/₃ (3x -2) – ¾ (2x -2)

9. Simplify by factorizing completely:

4y² – x²

2x² – yx -6y²

10. Simplify as far as possible.

3 – 1

x – y x + y

11. By calculation, find the coordinates of the intersection of the graphs y = x² + 2x -5

and y = 3x +1

12. Simplify:

(a) y² + 2y = ¼

y³ – y² – 6y

 (b) hence solve:- y₂ + 2y = ¼

 y³ – y² – 6y

13. A rectangular field measures 63.9m by 104.6metres find the minimum number of poles to be

erected for fencing if they are to be at most 2.4meters apart.

14. Factorize completely the expression

 75x² – 27y²

15. Every time an insect jumps forward the distance covered is half of the previous jump.

If the insect initially jumped 8.4cm, calculate

 (i) To the nearest two decimal places distance of the sixth jump

 (ii) The total distance covered after the sixth jump

16. Simplify P³ –Pq² + P²q – q³

P² + 2pq + q²

17. Simplify the expression:- 9x² – 4y²

12x² + yx – 6y²

18. Given that (x-3) (Ax²+bx+c) = x³-7x-6, find the value of A, B and C

19. a) solve for y in 8x(2²)ʸ=6×2ʸ-1

 b) Simplify completely 2x²-98 ÷ x+7

3x²-16x-35 3x+5

20. Simplify the expression.:

 4x² – y²

 2x² – 7xy + 3y²

21. Simplify P² -2Pq + q²

P³ –Pq²+ P²q – q³

22. The sum of two numbers is 15. The difference between five times the first number and

three times the second number is 19. Find the two numbers

23. Simplify the following expressions by reducing it to a single fraction

2x – 5 – 1 – x – x – 4

4 3 2

24. Simplify the expression:- 3a² + 4ab + b²

4a² + 3ab – b²

Algebraic expressions Answers

2. Let the daughter’s age 5yrs ago be x

Mother 4x

come;

Daughter = x + 9

Mother = 4x+ 9

4x + 9 = ⁵/₂ (x +9)

4x + 9 = 2.5x + 22.5

1.5x = 13.5

x = 9

Mother = 41yrs

14 + 41 =55

3. B.P = 160 x 50 = 24000

S.P = ((160 x 8) – (20 + 12)) x 180

8

= 28080

Profit = 28080 – 24000 = Shs.4080

4. a) 6a + 7a – 2b – 4b + 2

= 13a – 6b + 2

b) 2x – 2 – 3x + 2 = 2(2x – 2) – (3x + 2)

2x 4x 4x

= 4x – 3x – 4 – 2

4x

= x – 6

4x

5. 6u²y² + 13uy – 5 = (2uy + 5) (3xy – 1)

3u²y² – 13uy + X = (uy – 4) (3xy – 1)

(2xy +5) (3xy – 1)

(uy – 4) (3xy -1)

= 2xy + 5

Uy – 4

6. a) From x + y and x² = y² = 34

X = 8 – y

Substituting for x in x² – y² = 34

(8 – y) (8 – y) + y² = 34

64 – 8y – 8y + y² + y² = 34

64 – 16y + 2y² = 34

2y² – 16y + 64 – 34 = 0

2y² – 16y + 30 = 0

y² = 8y + 15 = 0

y (y – 3) – 5 (y-3) = 0 (y-5) (y – 3)

y is either 5 or 3

but x – y = 8

x is either 5 0r 3

 x² + 2xy + y² = 32 + 2 x 3 x 5 + 25

= 9 + 30 + 25 = 64

b) 2xy = 2 x 3 x 5 = 30

c) x² – 2xy + y² = 9 – 2 x 3 x 5 + 25 = 4

d) x = y = 8 and x² + y² = 34

x = 8 – y

 (8 – y)² + y² = 34

y² – 8y + 15 = 0

y² – 3y – 5y + 15 = 0

y(y -3) – 5(y – 3)

(y-3) = o y = 3

(y-5) = 0 y = 5

x + 3 = 8, x = 5 or x + 5 = 8

x = 3

 x is either 3 or 5

y is either 3 0r 5

7. 6x² + 35x – 6

2x² – 72

= 6x(x -+ 6) -1(x + 6)

2(x² – 36)

= (6x – 1) (x + 6)

2(x – 6) (x + 6)

 = 6x – 1

2(x – 6)

8^. 2/₅ (3x -2) – ¾ (2x -2)

= 8(3x-2) -9(2x-2)

12

= 24 x -16 – 18x + 18

= 6x + 2

12

= 2(3x +1)

12

= 3x + 1

6

9. Numerator:

4y² – x² = (2y +x) (2y – x)

Denominator :

2x² + 4yx + 3yx – 6y²

= (2x² – 4yx) + (3yx – 6y2)

= 2x(x-2y) + 3y(x-2y)

= (2xx+3y) (x-2y)

Combining : (2y + x) (2y-x)

(2x+3y) (x-2y)

– 2x + 3y or -2x -3y

2y + x 2y + x

10. 3(x + y) – (x –y)

X ² – y²

 = 3x + 3y – x + y

x² – y²

= 2(x +2y)

x² -y²

11. x² + 2x – 5 = 3x +1

x² – x – 6 – 6 =0

(x+2) (x-3) = 0

x =-2 or x = 3

When x =-2, y = 3x – 2 + 1 = -5 Point (-2, -5)

When x = 3, y = 3x x 3 +1 = 10 Point (3, 10)

12. (a) y(y + 2)

 y(y² – y -60

y(y+2) = y + 2

y(y² – y-6) (y+2) (y-3)

(b) y + 2 = ¼

(y + 2) (y-3)

4y + 8 = y² – y – 6

y² – 5y – 14 = 0

(y-7)(y+2)= 0

y=7

y=-2

13. 104.6 = 44 x 2

 2.4

63.9 = 26 x 2

 2.4

= 88 + 54= 142

14. 3 ( 25 x² – 9y²)

3 (5x + 3y) (5x – 3y)

15. i) d = 8.4 r = ½

 6^th jump = 8( ½ )^6-1

^8.4/₃₂

= 0.2625 = 0.26cm

ii) 56 = 9.4 (1 – ( ½ ) 6

1 – ½

= 8.4 x 63 x 2

64

= 16.54 cm

16. Factorizing the numerator

= p(p² – q²) + q(p²-q)

= (p+q) (p²-q²)

= (p+q) (p+q) n(p-q)

Factorising the denominator

(p+q) (p+q)

Numerator = p – q

Denominator

17. ( 3x + 2y ) ( 3x – 2y )

 ( 3x + 2y ) ( 3x – 2y )

3x + 2y

4x + 3y

18. (x – 3) (AX² +BX + C) = x³ – 7x – 6

AX³ + BX² +CX – 3AX² – 3BX – 3c = x³ – 7x – 6

A = 1

B – 3A = 0

B – 3 x 1 = 0

B = 3

-3c = -6

c = 2

19. a) 8(2²)^y = 6 x 2^y – 1

let t = 2^y

 8t² = 6t – 1

8t² – 4t – 2t + 1 = 0

(4t – 1) (2t – 1) = 0

t = ¼ or ½

t = 2^y = ¼ = 2^-2

y = -2

0r t = 2^y = ½ = 2^-1

y = -1

y = -2 0r -1

b) Numerator = 2x² – 98

= 2(x² – 49)

 = 2(x+ 7) (x -7)

Denominator = 3x² – 16x – 35

= 3x² – 21x + 5x – 35

= 3x(x -7) + 5(x – 7)

= (x – 7) (3x + 5)

2x² – 98

x + 7 = 2(x + 7) (x – 7) x (3x + 5)

3x² – 16x – 3 3x + 5 (3x + 5) (x – 7) (x + 7)

= 2

20. (2x- y) (2x + y) √

(x – 3y) (2x – y) √

2x + y √

x – 3y

21. P² – 2pq + q² = (p-q)²

P³ – pq² + p²q – q³

= p(p²-q²) + q (p² – q²)

= (p+ q) (p²-q²)

(p – q)²
= (p – q)² 

(p+ q) (P² – q²) (p+q)² (p-q)

= p- q

(p + q)²

22. Let the numbers be a and b

a + b = 15 – x3

5a – 3b = 19 x 1

 3a + 3b = 45

 5a – 3b = 19

8a = 64

a = 8

 b = 7

4 3 2

23. 3(2x-5) – 4(1-x) – 6(x-4)

12

6x – 15 – 4+ 4x – 6x + 24

12

4x – 5

12

24. 3a² + 4ab + b² = 3a² + 3ab + ab +b²

4a² + 3ab – b²
4a² + 4ab – ab – b²

= 3a(a+b) + b(a+b)

4a(a+b) – b(a+b)

= (3a+b) (a+b)

(a+b) (4a-b)

= 3a + b

4a -b