math doubts [NEW]

[zara]

Find the real roots of the equation 18/x⁴ + 1/x² =4

can summon plz explain?
                                          

[astarmathsandphysics]

Start by multiplying by x4. Home in 4 hours.

[slvri]

18/x⁴ + 1/x² =4
multiply by x⁴
18+x²=4x⁴
4x⁴-x²-18=0
4(x²)²-x²-18=0
u can see that this is a quadratic equation in x²
factorising we have
4x⁴-9x²+8x²-18=0
x²(4x²-9)+2(4x²-9)=0
(4x²-9)(x²+2)=0
either 4x²-9=0 or x²+2=0
4x²=9 or x²=-2
x²=9/4 (x²=-2 has no real roots, complex roots are +-sqrt(2)i)
so from the first one
x=sqrt(9/4)
x=+-(3/2)
hope this helped zara

[nid404]

 zara slvri has done it for ya and it's right....Thanks slvri

[zara]

18/x⁴ + 1/x² =4
multiply by x⁴
18+x²=4x⁴
4x⁴-x²-18=0
4(x²)²-x²-18=0
u can see that this is a quadratic equation in x²
factorising we have
4x⁴-9x²+8x²-18=0
x²(4x²-9)+2(4x²-9)=0
4(x²-9)(x²+2)=0
either 4x²-9=0 or x²+2=0
4x²=9 or x²=-2
x²=9/4 (x²=-2 has no real roots, complex roots are +-sqrt(2)i)
so from the first one
x=sqrt(9/4)
x=+-(3/2)





the ones in red i didnt get it....plz elaborate...
y is 4x²-9=0? wont it be x²-9=0?

[zara]

another question

The function h is defined by

h:x > 6x-x² for x>=3

Express 6x-x² in the form a-(x-b)², where a and b are positive constants.

[zara]

3) f is defined by:  f:x > 3x-2 for x E R

   g is defined by: g:x >6x-x² for x E R

Express gf(x) in terms of x, and hence show that the maximum value of gf(x) is 9.


i did the expression part...buh im confused in the second part...how to show it?
n yea one more thing....what does this mean "x E R"?
i don't remember our sir explaining this to us....n its der in our worksheet!

[Ghost Of Highbury]

ok as he is not online...i wud like to explain it to u..

the first one...u multiply it by x^4 so that u can simplify it and solve easily...as the divisor can be cancelled..

wen u multiply x^4 with 18/x^4 + 1/x^2 = u get => 18 (as the x^4 gets cancelled) + x^4/x^2 = 18 + x^2

next...

(4x^2 - 9)(x^2 + 2) = 0

this means that either (4x^2 - 9) = 0...or (x^2 +2) = 0....as something HAS to be multiplied by 0 to get a 0.

so 4x^2 - 9 = 0

4x^2 = 9..

and so on...

hope that helped

[Ghost Of Highbury]

x E R means that the values of x belong to the set of real values...


PS: wud try to solve thos.e

[zara]

thanks adi!!
tht helped!
n yea sure go ahead solve it!
its nt for specific people...whoever can is free to do! =)

[Ghost Of Highbury]

x^2 + 6x = 9 - (x-3)^2

therefore a = 9
             b = 3

[zara]

x^2 + 6x = 9 - (x-3)^2

therefore a = 9
             b = 3

im lost adi..:\
i need the steps if possible?!

[Ghost Of Highbury]

the general rule

x² + nx = (x + n/2)² - (n/2)²

n = 6

therefore n/2 = 3

here its  -x² + 6x = (x-3)² - (3)² = -x² + 6x = 9 - (x-3)²

[Ghost Of Highbury]

did u get g(f(x)) = -3x² +18x - 2?

[astarmathsandphysics]

the general rule

x² + nx = (x + n/2)² - (n/2)²

n = 6

therefore n/2 = 3

here its  -x² + 6x = (x-3)² - (3)² = -x² + 6x = (x-3)² - 9

Should be (x+3)²-9