Quadratic identies problem, that i dont understand (1 Viewer)

[wolf7]

i need help with 3 question that i dont understand?

first question

if x² = ax(ax-1)+bx for more than two values of xd, find a and b

second question

Find a, b and c if:

x² = a(x+1)²+bx+c

the equal(=) sign suppose to mean similar

third question
Find the values for A and B if:
4x²-12x + 9 = (Ax+B)²

in the thrid question, the equal sign also means similar.

Can some one help me with these question?

 

[wolf7]

plz, also can you show working out

 

[icycloud]

first question

if x² = ax(ax-1)+bx for more than two values of xd, find a and b

It can be proved that two quadratic equations which are equal for three or more distinct values of x (i.e. more than two values of x) agree for all values of x, and thus their coefficients are congruent (three horizontal lines). Thus,

LHS = x^2
RHS = ax^2 - ax + bx
= ax^2 + (b-a)x

Equating coefficients of LHS and RHS,

a = 1

b-a = 0
b-1=0
b = 1

Thus, a = 1, b = 1 #

second question

Find a, b and c if:

x² = a(x+1)²+bx+c

the equal(=) sign suppose to mean similar

Since the two equations are congruent (three horizontal lines), we can simply equate coefficients. There are two ways of doing this, one way is outlined below:

LHS = x^2
RHS = a[x^2 + 2x + 1] + bx + c
= ax^2 + 2ax + a + bx + c
= ax^2 + (2a+b)x + (a+c)

Thus, equating coefficients of LHS and RHS, we have:

a = 1

2a + b = 0,
b = -2(1)
= - 2

a+c = 0
c = -1

Thus, a = 1, b = -2, c = -1 #

third question
Find the values for A and B if:
4x²-12x + 9 = (Ax+B)²

in the thrid question, the equal sign also means similar.

As above, expand and equate coefficients. (The other way is to substitute various values of x to work out the coefficients. Sometimes this way is faster, it depends on the question.)

LHS = 4x^2 - 12x + 9
RHS = (Ax+B)^2
=(Ax)^2 + 2ABx + B^2
=A^2 x^2 + 2ABx + B^2

Equating coefficients,

A^2 = 4
A = +2 or -2

B^2 = 9
B = +3 or -3

2AB = -12
AB = -6
A = -6 / B

Thus,
A = 2 and B = -3 or
A = -2 and B = 3 #

 

[wolf7]

thanks, for the help, made it sound so easy