Quadratic Equations Set – 8

1. I. 2x2 – 15√3x + 84 = 0
II. 3y2 – 10√3y + 9 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option A
Solution: 

2x2 – 15√3x + 84 = 0
Now multiply 2 and 84 = 168
we have √3 in equation, so divide, 168/3 = 56
Now make factors so as by multiply you get 56, and by addition or subtraction you get –15
we have factors (-8) and (-7)
So 2x2 – 15√3x + 84 = 0
gives
2x2 – 8√3x – 7√3x + 84 = 0
2x (x – 4√3) – 7√3 (x – 4√3x) = 0
So x = 7√3/2, 4√3
Similarly for
3y2 – 10√3y + 9 = 0
Multiply 3 and 9 = 27
we have √3 in equation, so divide, 27/3 = 9
Now make factors so as by multiply you get 9, and by addition or subtraction you get –10
we have factors (-9) and (-1)
So 3y2 – 10√3y + 9 = 0
gives
3y2 – 9√3y – √3y + 9 = 0
3x (x – 3√3) – √3 (x – 3√3x) = 0
Put all values on number line and analyze the relationship
√3/3 …. 3√3 ….. 7√3/2 …… 4√3

 

2. I. x2 + √5x – 10 = 0
II. 2y2 + 9√5y + 50 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option C
Solution: 

x2 + √5x – 10 = 0
x2 + 2√5x – √5x – 10 = 0
Gives x = -2√5, √5
2y2 + 9√5y + 50 = 0
2y2 + 4√5y + 5√5y + 50 = 0
Gives y = -2√5, -5√5/2
Put all values on number line and analyze the relationship
-5√5/2….. -2√5….. √5

 

3. I. 2x2 – (8+√3)x + 4√3 = 0
II. 3y2 – (6+2√3)y + 4√3 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option E
Solution: 

2x2 – (8+√3)x + 4√3 = 0
By multiplying we have to 2*4√3 = 8√3 and by adding/subtracting we have to get – (8+√3)
So factors are -8 and -√3
So 2x2 – (8+√3)x + 4√3 = 0
Gives
2x2 – 8x – √3x + 4√3 = 0
2x(x- 4) – √3(x – 4) = 0
So x = 4, √3/2
NEXT
3y2 – (6+2√3)y + 4√3 = 0
By multiplying we have to 3*4√3 = 12√3 and by adding/subtracting we have to get –(6+2√3)
So factors are -6 and -2√3
So 3y2 – (6+2√3)y + 4√3 = 0
Gives
3y2 – 6y – 2√3y + 4√3 = 0
3y(y- 2) – 2√3(y – 2) = 0
So x = 2, 2√3/3
Put all values on number line and analyze the relationship
√3/2…… 2√3/3…… 2… 4

 

4. I. x2 – (2+√5)x + 2√5 = 0
II. 2y2 – (6+3√5)y + 9√5 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option B
Solution: 

x2 – (2+√5)x + 2√5 = 0
By multiplying we have to 2√5 and by adding/subtracting we have to get – (2+√5)
So factors are -2 and -√5
So x2 – (2+√5)x + 2√5 = 0
Gives
x2 – 2x – √5x + 2√5 = 0
x(x- 2) – √5(x – 2) = 0
So x = 2, √5
NEXT
2y2 – (6+3√5)y + 9√5 = 0
By multiplying we have to 2*9√5 = 18√5 and by adding/subtracting we have to get –(6+3√5)
So factors are -6 and -3√5
So 2y2 – (6+3√5)y + 9√5 = 0
Gives
2y2 – 6y – 3√5y + 9√5 = 0
2y(y- 3) – 3√5(y – 3) = 0
So x = 3, 3√5/2
Put all values on number line and analyze the relationship
2…… √5…… 3… 3√5/2

 

5. I. 3x2 + 5√2x – 24 = 0
II. y2 – 6√2y + 16 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option B
Solution: 

3x2 + 5√2x – 24 = 0
3x2 + 9√2x – 4√2x – 24 = 0
Gives x = -3√2, 4√2/3
y2 – 6√2y + 16 = 0
y2 – 2√2y – 4√2y + 16 = 0
Gives y = 2√2, 4√2
Put all values on number line and analyze the relationship
3√2……. 4√2/3…… 2√2….. 4√2

 

6. I. 3x2 – 23x + 40 = 0
II. 3y2 – 8y + 4 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option A
Solution: 

3x2 – 23x + 40 = 0
3x2 – 15x – 8x + 40 = 0
Gives x = 5, 8/3
3y2 – 8y + 4 = 0
3y2 – 6y – 2y + 4 = 0
Gives y = 2/3, 2
Put all values on number line and analyze the relationship
2/3….. 2….. 8/3….. 5

 

7. I. 5x2 – 17x + 6 = 0
II. 4y2 – 16y + 7 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option E
Solution: 

5x2 – 17x + 6 = 0
5x2 – 15x – 2x + 6 = 0
Gives x = 2/5, 3
4y2 – 16y + 7 = 0
4y2 – 2y – 14y + 7 = 0
Gives y = 1/2, 7/2
Put all values on number line and analyze the relationship
2/5….. 1/2….. 3…. 7/2

 

8. I. 3x2 – 14x + 8 = 0
II. 3y2 – 20y + 12 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option E
Solution: 

3x2 – 14x + 8 = 0
3x2 – 12x – 2x + 8 = 0
Gives x = 4, 2/3
3y2 – 20y + 12 = 0
3y2 – 18y – 2y + 12 = 0
Gives y = 2/3, 6
Put all values on number line and analyze the relationship
2/3…….. 4….. 6

 

9. I. 12x2 + 25x + 12 = 0
II. 3y2 + 22y + 24 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option C
Solution: 

12x2 + 25x + 12 = 0
12x2 + 16x + 9x + 12 = 0
Gives x = -4/3, -3/4
3y2 + 22y + 24 = 0
3y2 + 18y + 4y + 24 = 0
Gives y = -4/3, -6
Put all values on number line and analyze the relationship
-6…… -4/3…… -3/4

 

10. I. 6x2 + x – 2 = 0
II. 3y2 – 22y + 40 = 0
A) x > y
B) x < y
C) x ≥ y
D) x ≤ y
E) x = y or relation cannot be established

Answer & Explanation

 Option B
Solution: 

6x2 + x – 2 = 0
6x2 + 4x – 3x – 2 = 0
Gives x = 1/2, -2/3
3y2 – 22y + 40 = 0
3y2 – 12y – 10y + 40 = 0
Gives y = 10/3, 4
Put all values on number line and analyze the relationship
-2/3…… 1/2…… 10/3….. 4

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