Question

Solve the following problems in the attachment with complete steps …

Points : 40 ☺

Answer 1

1) 2x² + 7/2x + 3/4

coefficient of x² = 2
coefficient of x = 7/2
constant = 3/4
maximum power of x =2
hence degree =2

now,
2x² + 7/2x + 3/4
let two zeros A and B here

then,
A + B = -coefficient of x/coefficient of x² = - (7/2) /(2) = -7/4

A.B = constant/coefficient of x²
=( 3/4)/(2) = 3/2

use quadratic formula,
x ={-b ±√(b²-4ac)}/2

={ -7/2 ±√(49/4 -24/4)}/2
={ -7/2 ±5/2 }/2 = -3, -1/2

2) x² + 1/6x -2

coefficient of x² = 1
coefficient of x = 1/6
constant = -2

let A and B are zeros of polynomial then,
A + B = - (1/6)/1 = - 1/6

A.B = (-2)/1 = -2

x ={ -1/6±√(1/36+8)}/2
={-1/6± √(289/36)}/2
=(-1/6±17/6)/2
= -3/2 , 4/3

3)4√3x² + 5x -2√3

coefficient of x² = 4√3
coefficient of x = 5
constant = -2√3

let A and B are the zeros of this polynomial then

A + B = -(5/4√3) = -5/4√3

A.B = (-2√3)/(4√3) = -1/2

x ={-5±√(25+96)}/8√3
={-5±11}/8√3
= -2/√3 , √3/4

4) 4x² + 5√2x -3

coefficient of x² = 4
coefficient of x = 5√2
constant = -3

let A and B are the zeros of polynomial then,

A + B = -(5√2)/4 = -5/2√ 2

A.B = ( -3)/4 = -3/4

x = {-5√2±√(50+48)}/8
={-5√2±7√2}/8
= -3/√2 , 1/2√2

Answer 2

1)let p(x) = 2x²-7/2 x +3/4
to get zeroes we will take p(x) =0

2x² -7/2 x+3/4 =0
multiply each term with 4
8x²-14x+3=0
8x² -12x-2x+3=0
4x(2x-3) -1(2x-3) =0
(2x-3) (4x-1) =0
∴2x-3 =0 or 4x-1 =0
x= 3/2 or x= 1/4
compare  p(x) with ax²+bx +c 
a= 2 , b= -7/2 , c= 3/4

i) sum of the zeroes= 3/2 +1/4 = (6+1)/4 =7/4 :;
-b/a = - (-7/2) / 2 = 7/4

ii) product of the zeroes = (3/2) (1/4) =3/8 ;
 c/a = (3/4)/2 = 3/8

2) let g(x) = x²+x/6-2
g(x) =0
x²+x/6 -2 =0
multiply each term with 6
6x²+x -12=0
⇒6x²+9x-8x -12=0
⇒3x(2x+3) -4(2x+3)=0
(2x+3)(3x-4) =0
∴2x+3=0 or 3x-4=0
x= -3/2 or x= 4/3
α= -3/2 , β= 4/3
i) α+β= -3/2 +4/3 = (-9+8)/6= -1/6
α+β = -b/a = - (1/6) / 1 = -1/6
ii) αβ = (-3/2 ) (4/3) = -2
αβ= c/a = (-2)/1 =-2

3) let h(x) = 4√3 x² +5x-2√3
h(x) =0
4√3 x²+5x-2√3=0
⇒4√3x² +8x -3x -2√3=0
⇒4x(√3x+2) -√3( √3x+2)=0
⇒(√3x+2)(4x- √3)=0
∴√3x+2=0 or 4x-√3 =0
x= 2/√3 or x= √3/4
α= -2/√3 or β = √3/4
i) α+β =- 2/√3 + √3/4 = (-8+3)/(4√3) = -5/(4√3)
α+β = - b/a= = - 5/(4√3) 
ii) αβ = (-2/√3)(√3/4)= -1/2
αβ= c/a= (-2√3)/(4√3)= -1/2
4)
let p(x) = 4x²+√2x-3
p(x) =0
4x²+√2x -3 =0
⇒4x² +3√2x -2√2x-3=0
⇒√2x (2√2x+3) -(2√2x+3)=0
⇒(2√2x+3)(√2x-1)=0
∴2√2x+3=0 or √2x-1 =0
x=(- 3)/(2√2) or x= 1/√2
α=- 3/(2√2)  or β= 1/√2
i) α+β =- 3/(2√2) + 1/√2 = (-3+2) /(2√2) = -1/(2√2)
α+β = -b/a = - (√2) /4= (-√2) / (2*√2*√2) = -1/2√2
ii) αβ = (-3/(2√2) (1/√2) = (-3) /4
αβ = c/a= (-3)/4