Find the value of a/b + b/a, if a and b are the roots of the quadratic equation x^{2} + 8x + 4 = 0?
Explanation:
a/b + b/a = (a^{2} + b^{2})/ab = (a^{2} + b^{2} + a + b)/ab
= [(a + b)^{2}  2ab]/ab
a + b = 8/1 = 8
ab = 4/1 = 4
Hence a/b + b/a = [(8)^{2}  2(4)]/4 = 56/4 = 14.
Find the quadratic equations whose roots are the reciprocals of the roots of 2x^{2} + 5x + 3 = 0?
Explanation:
The quadratic equation whose roots are reciprocal of 2x^{2} + 5x + 3 = 0 can be obtained by replacing x by 1/x.
Hence, 2(1/x)^{2} + 5(1/x) + 3 = 0
=> 3x^{2} + 5x + 2 = 0
A man could buy a certain number of notebooks for Rs.300. If each notebook cost is Rs.5 more, he could have bought 10 notebooks less for the same amount. Find the price of each notebook?
Explanation:
Let the price of each note book be Rs.x.
Let the number of note books which can be brought for Rs.300 each at a price of Rs.x be y.
Hence xy = 300
=> y = 300/x
(x + 5)(y  10) = 300 => xy + 5y  10x  50 = xy
=>5(300/x)  10x  50 = 0 => 150 + x^{2} + 5x = 0
multiplying both sides by 1/10x
=> x^{2} + 15x  10x  150 = 0
=> x(x + 15)  10(x + 15) = 0
=> x = 10 or 15
As x>0, x = 10.
I. a^{2}  7a + 12 = 0,
II. b^{2}  3b + 2 = 0 to solve both the equations to find the values of a and b?
Explanation:
I.(a  3)(a  4) = 0
=> a = 3, 4
II. (b  2)(b  1) = 0
=> b = 1, 2
=> a > b
I. a^{2}  9a + 20 = 0,
II. 2b^{2}  5b  12 = 0 to solve both the equations to find the values of a and b?
Explanation:
I. (a  5)(a  4) = 0
=> a = 5, 4
II. (2b + 3)(b  4) = 0
=> b = 4, 3/2 => a ≥ b
I. a^{2} + 11a + 30 = 0,
II. b^{2} + 6b + 5 = 0 to solve both the equations to find the values of a and b?
Explanation:
I. (a + 6)(a + 5) = 0
=> a = 6, 5
II. (b + 5)(b + 1) = 0
=> b = 5, 1 => a ≤ b
I. a^{2} + 8a + 16 = 0,
II. b^{2}  4b + 3 = 0 to solve both the equations to find the values of a and b?
Explanation:
I. (a + 4)^{2} = 0 => a = 4
II.(b  3)(b  1) = 0
=> b = 1, 3 => a < b
I. a^{2}  2a  8 = 0,
II. b^{2} = 9 to solve both the equations to find the values of a and b?
Explanation:
I. (a  4)(a + 2) = 0
=> a = 4, 2
II. b^{2} = 9
=> b = ± 3
2 < 3, 2 > 3, 4 > 3, 4 > 3,
No relation can be established between a and b.
I. x^{2} + 5x + 6 = 0,
II. y^{2} + 9y +14 = 0 to solve both the equations to find the values of x and y?
I. x2 + 3x + 2x + 6 = 0
=> (x + 3)(x + 2) = 0 => x = 3 or 2
II. y2 + 7y + 2y + 14 = 0
=> (y + 7)(y + 2) = 0 => y = 7 or 2
No relationship can be established between x and y.
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