Question

factor completely
(5x^4-44x^2+63)

Answer 1

X= +-3/sqrt(5), x=+-sqrt(7)

Answer 2

Make the substitution t = x^2

The problem will now become a quadratic expression in t. See if you can factor it. Later put back x^2 in pace of t and see if it can be factored further.

Answer 3

Yup sub in something for x^2

Answer 4

huh

Answer 5

im confused

Answer 6

Let t = x^2
5x^4-44x^2+63 becomes 5t^2 - 44t + 63
Now this is a quadratic equation in t. See if it can be factored.

Answer 7

u multiply 5 n 63 right

Answer 8

You can try this:
5t^2 - 44t + 63 = 5t^2 - 35t - 9t + 63
Now try factoring it.

Answer 9

wat do u do after that thats the point im at n dont understand it

Answer 10

5t^2 - 44t + 63 = 5t^2 - 35t - 9t + 63
= 5t(t - 7) - 9(t - 7)
= (5t - 9)(t - 7)
Now put back t = x^2
(5x^2 - 9)(x^2 - 7)

Do you know if they allow you to use radicals in factors? If allowed we can factor this further.

Answer 11

wats radicals

Answer 12

symbols such as the square root sign, cube root, etc.

Answer 13

idk wat that is

Answer 14

If you look in the chapter that asked this question and you see some examples where they have factored and radical symbols appear in those factors then we can do the same here and factor further.

Answer 15

o yea i think u do have to do that

Answer 16

Okay in that case let me introduce radical symbols:

There is an identity for difference of two squares:
a^2 - b^2 = (a + b)(a - b)

So we need to put each factor in \[\Large (5x^2 - 9)(x^2 - 7) = ((\sqrt{5}x)^2 - 3^2)(x^2 - \sqrt{7}^2)\]as a difference of two squares as shown:

Answer 17

huh

Answer 18

And the answer will be: \[\Large (\sqrt{5}x + 3)(\sqrt{5}x - 3)(x + \sqrt{7})(x - \sqrt{7})\]

Answer 19

okay, we have this identity called difference of two squares:
a^2 - b^2 = (a + b)(a - b)

So if we can put something that looks like the expression on the left then we can factor it the way it appears on the right.

Just to give you an example: If we see (x^2 - 4) we can write is as: (x^2 - 2^2)
Now this is a difference of two squares. Compare this to a^2 - b^2 and you can see x is a and 2 is b. So it can be factored as: (a + b)(a - b) or (x + 2)(x - 2).
So (x^2 - 2^2) = (x + 2)(x - 2)

Follow so far?

Answer 20

yea

Answer 21

So we have two factors: (5x^2 - 9) and (x^2 - 7)
Let us take the first factor (5x^2 - 9). I want to make this as a difference of two squares. So you take the square root of the first term and then square it. You take the square root of the second term and then square it. Square root and squares will cancel out and so we won't be changing the expression. It is like writing 16 as square of (square root of 16). The square and square root will cancel out leaving with just 16 as before.

So 5x^2 can be written as square of (square root of 5x^2).
square root of 5x^2 = square root of 5 * square root of x^2 = \[\sqrt{5} * x\]

Answer 22

9 can be written as square of (square root of 9) = square of (3)
Therefore, \[(5x^2 - 9) = (\sqrt{5}x)^2 - 3^2 = (\sqrt{5}x + 3)(\sqrt{5}x - 3)\]

Answer 23

ok

Answer 24

\[(x^2 - 7) = x^2 - (\sqrt{7})^2 = (x + \sqrt{7})(x - \sqrt{7})\]

Answer 25

ok

Answer 26

\[\Large (\sqrt{5}x + 3)(\sqrt{5}x - 3)(x + \sqrt{7})(x - \sqrt{7})\]is the result when we factor the original problem completely.

Answer 27

ok thanks

Answer 28

you are welcome.

Answer 29

this is easy after u gt the hang of it