9x^4=25x^2-16
solve using an appropriate substitution

Question

9x^4=25x^2-16
solve using an appropriate substitution

jim_thompson5910 · Answer

Let z = x^2

So z^2 = (x^2)^2 = x^4

This means

9x^4=25x^2-16

becomes

9z^2=25z-16

jim_thompson5910 · Answer

Solve that last equation for z. Then use the solutions in terms of z to find the solutions in terms of x.

anonymous · Answer

okay but heres where im stuck:  9z^2-25y+16=0.. how do i factor that?

jim_thompson5910 · Answer

Find two numbers that multiply to 9*16 = 144 AND add to -25

anonymous · Answer

multiply 9 and 16?

jim_thompson5910 · Answer

yes that gives you 144

jim_thompson5910 · Answer

you need to find two numbers that multiply to 144 and add to -25

anonymous · Answer

i thought we were finding two numbers that multiplied to equal 16 and which add to -25?

jim_thompson5910 · Answer

that only works if the leading coefficient is 1

anonymous · Answer

oh really now

anonymous · Answer

so in any other case you are multiplying the first and last part?

jim_thompson5910 · Answer

yes given ax^2+bx+c, it factors when you can find two numbers that multiply to ac and add to b

jim_thompson5910 · Answer

example:

Factor x^2+5x+6

that's really 1x^2+5x+6

So we need to find two numbers that multiply to 1*6 = 6 and add to 5...those numbers are 2 and 3

So it factors to (x+2)(x+3)

Note: this last jump only works if the leading coefficient is 1

anonymous · Answer

ok so if it was 2x^2+5x+6 youd be looking for what times what equals 12 and adds to 5?

jim_thompson5910 · Answer

you got it

anonymous · Answer

okay so its gonna be -9 and -16

jim_thompson5910 · Answer

good, now you can replace -25z with -9z-16z to go from

9z^2-25z+16=0

to

9z^2-9z-16z+16=0

jim_thompson5910 · Answer

Then you factor by grouping

anonymous · Answer

(9y-16)(y-1)

jim_thompson5910 · Answer

good

anonymous · Answer

now i need to convert back to x's?

anonymous · Answer

nailed it. nice

jim_thompson5910 · Answer

you can if you want, doing so will turn 

(9y-16)(y-1)

into

(9x^2-16)(x^2-1)

assuming you let y = x^2

jim_thompson5910 · Answer

Then you can factor each piece separately

anonymous · Answer

ya they expect me to. my answer matches the books now

jim_thompson5910 · Answer

alright great

anonymous · Answer

perfect. thanks so much for the help

jim_thompson5910 · Answer

you're welcome

anonymous · Answer

that little detail about a*c was a big break through for me

jim_thompson5910 · Answer

I'm glad I could shed light on it