Simplify 2/(3x^2-20x+32)+1/(3x^2-14x+16)

Math
23×2-20x+32+13×2-14x+16
Simplify each term.
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Factor by grouping.
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For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=3⋅32=96 and whose sum is b=-20.
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Factor -20 out of -20x.
23×2-20(x)+32+13×2-14x+16
Rewrite -20 as -8 plus -12
23×2+(-8-12)x+32+13×2-14x+16
Apply the distributive property.
23×2-8x-12x+32+13×2-14x+16
23×2-8x-12x+32+13×2-14x+16
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
2(3×2-8x)-12x+32+13×2-14x+16
Factor out the greatest common factor (GCF) from each group.
2x(3x-8)-4(3x-8)+13×2-14x+16
2x(3x-8)-4(3x-8)+13×2-14x+16
Factor the polynomial by factoring out the greatest common factor, 3x-8.
2(3x-8)(x-4)+13×2-14x+16
2(3x-8)(x-4)+13×2-14x+16
Factor by grouping.
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For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=3⋅16=48 and whose sum is b=-14.
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Factor -14 out of -14x.
2(3x-8)(x-4)+13×2-14(x)+16
Rewrite -14 as -6 plus -8
2(3x-8)(x-4)+13×2+(-6-8)x+16
Apply the distributive property.
2(3x-8)(x-4)+13×2-6x-8x+16
2(3x-8)(x-4)+13×2-6x-8x+16
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
2(3x-8)(x-4)+1(3×2-6x)-8x+16
Factor out the greatest common factor (GCF) from each group.
2(3x-8)(x-4)+13x(x-2)-8(x-2)
2(3x-8)(x-4)+13x(x-2)-8(x-2)
Factor the polynomial by factoring out the greatest common factor, x-2.
2(3x-8)(x-4)+1(x-2)(3x-8)
2(3x-8)(x-4)+1(x-2)(3x-8)
2(3x-8)(x-4)+1(x-2)(3x-8)
To write 2(3x-8)(x-4) as a fraction with a common denominator, multiply by x-2x-2.
2(3x-8)(x-4)⋅x-2x-2+1(x-2)(3x-8)
To write 1(x-2)(3x-8) as a fraction with a common denominator, multiply by x-4x-4.
2(3x-8)(x-4)⋅x-2x-2+1(x-2)(3x-8)⋅x-4x-4
Write each expression with a common denominator of (3x-8)(x-4)(x-2), by multiplying each by an appropriate factor of 1.
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Multiply 2(3x-8)(x-4) and x-2x-2.
2(x-2)(3x-8)(x-4)(x-2)+1(x-2)(3x-8)⋅x-4x-4
Multiply 1(x-2)(3x-8) and x-4x-4.
2(x-2)(3x-8)(x-4)(x-2)+x-4(x-2)(3x-8)(x-4)
Reorder the factors of (3x-8)(x-4)(x-2).
2(x-2)(3x-8)(x-2)(x-4)+x-4(x-2)(3x-8)(x-4)
Reorder the factors of (x-2)(3x-8)(x-4).
2(x-2)(3x-8)(x-2)(x-4)+x-4(3x-8)(x-2)(x-4)
2(x-2)(3x-8)(x-2)(x-4)+x-4(3x-8)(x-2)(x-4)
Combine the numerators over the common denominator.
2(x-2)+x-4(3x-8)(x-2)(x-4)
Simplify the numerator.
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Apply the distributive property.
2x+2⋅-2+x-4(3x-8)(x-2)(x-4)
Multiply 2 by -2.
2x-4+x-4(3x-8)(x-2)(x-4)
Add 2x and x.
3x-4-4(3x-8)(x-2)(x-4)
Subtract 4 from -4.
3x-8(3x-8)(x-2)(x-4)
3x-8(3x-8)(x-2)(x-4)
Cancel the common factor of 3x-8.
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Cancel the common factor.
3x-8(3x-8)(x-2)(x-4)
Rewrite the expression.
1(x-2)(x-4)
1(x-2)(x-4)
Simplify 2/(3x^2-20x+32)+1/(3x^2-14x+16)

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