急!急!急! Quadratic Equation F4.

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Q 1 : Find the value of p if one root of 3x^2-4x+p=0 is three times the other.

A 1 :

3x^2-4x+p=0
(3x-√p)(x-√p) = 0 <--- ( one root of 3x^2-4x+p=0 is three times the other )
4x√p = 4x
√p = 1
p = 1

Answer : p = 1 .

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Q 2 : If the roots 3x^2+(k+3)x-12=0 are numerically equal but opposite in sign, find the roots and the value of k.

A 2 :

3x^2+(k+3)x-12=0
(x√3+√12)(x√3-√12)=0 <--- ( the roots 3x^2+(k+3)x-12=0 are numerically equal but opposite in sign )
k + 3 = 6x-6x
k = -3

Answer : k = -3 .

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Q 3 : If one root of the equation ax^2-10x+(2a-1)=0 is the reciprocal of the other, find the value of a.

A 3 :

ax^2-10x+(2a-1)=0
x^2-10x/a+(2a-1)/a=0
(x-1/k)(x-k)=0

1 = (2a-1)/a
a = 2a - 1
a = 1

Answer : a = 1 .

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Q 4a : If the different between the roots of 2x^2-6x+c=0 is 5, find the roots .

A 4a :

2x^2-6x+c=0
x^2-3x+c/2=0
(x-k)[x-(k+5)]=0 <--- ( the different between the roots of 2x^2-6x+c=0 is 5 )

2k+5 = 3
2k = -2
k = -1

k+5 = 4

Answer : The two roots are -1 and 4 .

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Q 4b : If the different between the roots of 2x^2-6x+c=0 is 5, find the value of c .

In Q 4a , we know 2x^2-6x+c ≡ (x+1)(x-4)

2x^2-6x+c ≡ (x+1)(x-4)
2x^2-6x+c ≡ x^2-3x-4
2x^2-6x+c ≡ 2x^2-6x-8
c = 8

Answer : c = 8.

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