CHAPTER ONE: QUADRATICS

Tutorial 1

1. Multiply out the following expressions and collect like terms.

a. (a+2)(a+3)

b. (c−4)(c−2)

c. (e+6)(e−1)

d. (h+5 )2

Answer: a. a2+5 a+6 b. c2−6 c+8 c.

e2+5e-6 d. h+10h+25

2. Factorise the following quadratic expressions.

a. x2+6 x+8

b. y2+9 y+20

c. a2−9

d. r2−2r−15

e. ( x+3 )2−9

f. 2 x2+5 x+2

g. 2 x2+14 x+24

h. 9 x2−6 x+1

Answer: a. ( x+2 ) (x+4 ) b. ( y+4 ) ( y+5 ) c. (r−5 ) (r+3 ) d. (a+3 ) (a−3 ) e. x (x+6 )

f. (2 x+1 ) ( x+2 ) g. 2 ( x+3 ) ( x+4 ) h.

(3 x−1 )2

3. Solve the following equations.

a. x2−11 x+24=0

b. x2−64=0

c. 3 x2−5 x+2=0

d. 9 x2−12 x+4=0

e. x2−x=20

f. x−1=6x

Answer: a. x=8 or x=3 b. x=−8 or x=8

c. x=2/3 or x=1 d. x=23

(repeated)

e. x=−5 or x=4 f. x=−2 or x=3

4. Write the following in completed square form.

a. 2 x2+4 x+6

b. −1 x2−2x+5

c. 5 x2−10 x+7

d. −3 x2−12x

Answer: a. 2 ( x+1 )2+4 b. −( x+1 )2+6 c.

5 ( x−1 )2+2 d. −3 ( x+2 )2+12

5. Use the quadratic formula to solve the following equations, where possible.

a. x2+8 x+5=0

b. x2−5 x−19=0

c. 3 x2+2x−4=0

d. x2−12=0

Answer: a. x=−0.683 or x=−7.317b. x=7.525 or x=−2.525c. x=0.869 or x=−1.535d. x=3.464 or x=−3.464

6. Find the value of discriminant and use it to find the number of real roots for each of the following equations.

a. x2−3 x+4=0

b. 4 x2−3x=0

c. 3 x2+4 x+1=0

d. x2+10x+25=0

Answer: a. -7, no real roots b. 9, two real rootsc. 4, two real roots d. 0, one repeated root

7. Sketch the quadratic curve.

a. y=x2+4 x+9

b. y=x2+4 x+3

c. y=x2+6 x−1

d. y=x2+x+2

e. y=x2+ 12x+1

Answer: a. ( x+2 )2+5; x = -2; (-2, 5) intercept-y (0, 9)

b. ( x+2 )2−1; x = -2; (-2, -1) intercept-y (0, 3)

c. ( x+3 )2−10; x = -3; (-3, -10) intercept-y (0, -1)

d. (x+ 12 )

2

+ 74

; x = −12

; (−12

, 74

) intercept-y (0,

2)

e. (x−14 )

2

+ 1516

; x = 14

; (14

, 1516

) intercept-y (0, 1)