How do you simplify #(6x^2-54x+84)/(8x^2-40x+48) div (x^2+x-56)/(x^2+12x+32)#?

Answer 1

#E = 3/4 * (x+4)/(x-3)#

Start by writing down your starting expression

#E = (6x^2 - 54x + 84)/(8x^2 - 40x + 48) * (x^2 + 12x + 32)/(x^2 + x - 56)#
For the first fraction, you can factor the numerator by #6# and the denominator by #8# to get
#E = (6 * (x^2 - 9x + 14))/(8 * (x^2 - 5x + 6)) * (x^2 + 12x + 32)/(x^2 + x - 56)#

All these quadratics can be easily factored by using the sum/product technique to get

#x^2 - 9x + 14 = x^2 - 2x - 7x + 14 = (x-2)(x-7)#
#x^2 - 5x + 6 = x^2 - 2x - 3x + 6 = (x-2)(x-3)#
#x^2 + 12x + 32 = x^2 + 4x + 8x + 32 = (x+ 8)(x + 4)#
#x^2 + x - 56 = x^2 + 8x - 7x + 56 = (x+8)(x-7)#

The expression will thus be equal to

#E = (6 * color(red)(cancel(color(black)((x-2)))) * color(green)(cancel(color(black)((x-7)))))/(8 * color(red)(cancel(color(black)((x-2)))) * (x-3)) * (color(blue)(cancel(color(black)((x+8)))) * (x+4))/(color(blue)(cancel(color(black)((x+8)))) * color(green)(cancel(color(black)((x-7)))))#
#E = color(green)(3/4 * (x+4)/(x-3)#
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Answer 2

To simplify the expression (6x^2-54x+84)/(8x^2-40x+48) divided by (x^2+x-56)/(x^2+12x+32), we can first factorize the numerator and denominator of both fractions:

(6x^2-54x+84) factors to 6(x-2)(x-7) (8x^2-40x+48) factors to 8(x-2)(x-3) (x^2+x-56) factors to (x-7)(x+8) (x^2+12x+32) factors to (x+4)(x+8)

Next, we can rewrite the expression as a multiplication by flipping the second fraction and multiplying:

(6(x-2)(x-7))/(8(x-2)(x-3)) * ((x+8)(x-7))/(x+4)(x+8)

Now, we can cancel out common factors:

(6)/(8(x-3)) * 1/(x+4)

Simplifying further:

3/(4(x-3)(x+4))

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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