Respuesta :
Answer:
[tex]x=\frac{9}{d-1},\:P=\frac{-297d+378}{\left(d-1\right)^2}+343[/tex]
Step-by-step explanation:
Let us start by isolating x for dx = x + 9.
dx - x = x + 9 - x > dx - x = 9.
Factor out the common term of x > x(d - 1) = 9.
Now divide both sides by d - 1 > [tex]\frac{x\left(d-1\right)}{d-1}=\frac{9}{d-1};\quad \:d\ne \:1[/tex]. Go ahead and simplify.
[tex]x=\frac{9}{d-1};\quad \:d\ne \:1[/tex].
Now, [tex]\mathrm{For\:}P=x+1x^2-34x+343, \mathrm{Subsititute\:}x=\frac{9}{d-1}[/tex].
[tex]P=\frac{9}{d-1}+1\cdot \left(\frac{9}{d-1}\right)^2-34\cdot \frac{9}{d-1}+343[/tex].
Group the like terms... [tex]1\cdot \left(\frac{9}{d-1}\right)^2+\frac{9}{d-1}-34\cdot \frac{9}{d-1}+343[/tex].
[tex]\mathrm{Add\:similar\:elements:}\:\frac{9}{d-1}-34\cdot \frac{9}{d-1}=-33\cdot \frac{9}{d-1}[/tex] > [tex]1\cdot \left(\frac{9}{d-1}\right)^2-33\cdot \frac{9}{d-1}+343[/tex].
Now for [tex]1\cdot \left(\frac{9}{d-1}\right)^2 > \mathrm{Apply\:exponent\:rule}: \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c} > \frac{9^2}{\left(d-1\right)^2} = 1\cdot \frac{9^2}{\left(d-1\right)^2}[/tex].
[tex]\mathrm{Multiply:}\:1\cdot \frac{9^2}{\left(d-1\right)^2}=\frac{9^2}{\left(d-1\right)^2}[/tex].
Now for [tex]33\cdot \frac{9}{d-1} > \mathrm{Multiply\:fractions}: \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c} > \frac{9\cdot \:33}{d-1} > \frac{297}{d-1}[/tex].
Thus we then get [tex]\frac{9^2}{\left(d-1\right)^2}-\frac{297}{d-1}+343[/tex].
Now we want to combine fractions. [tex]\frac{9^2}{\left(d-1\right)^2}-\frac{297}{d-1}[/tex].
[tex]\mathrm{Compute\:an\:expression\:comprised\:of\:factors\:that\:appear\:either\:in\:}\left(d-1\right)^2\mathrm{\:or\:}d-1 > This\: is \:the\:LCM > \left(d-1\right)^2[/tex]
[tex]\mathrm{For}\:\frac{297}{d-1}:\:\mathrm{multiply\:the\:denominator\:and\:numerator\:by\:}\:d-1 > \frac{297}{d-1}=\frac{297\left(d-1\right)}{\left(d-1\right)\left(d-1\right)}=\frac{297\left(d-1\right)}{\left(d-1\right)^2}[/tex]
[tex]\frac{9^2}{\left(d-1\right)^2}-\frac{297\left(d-1\right)}{\left(d-1\right)^2} > \mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}> \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}[/tex]
[tex]\frac{9^2-297\left(d-1\right)}{\left(d-1\right)^2} > 9^2=81 > \frac{81-297\left(d-1\right)}{\left(d-1\right)^2}[/tex].
Expand [tex]81-297\left(d-1\right) > -297\left(d-1\right) > \mathrm{Apply\:the\:distributive\:law}: \:a\left(b-c\right)=ab-ac[/tex].
[tex]-297d-\left(-297\right)\cdot \:1 > \mathrm{Apply\:minus-plus\:rules} > -\left(-a\right)=a > -297d+297\cdot \:1[/tex].
[tex]\mathrm{Multiply\:the\:numbers:}\:297\cdot \:1=297 > -297d+297 > 81-297d+297 > \mathrm{Add\:the\:numbers:}\:81+297=378 > -297d+378 > \frac{-297d+378}{\left(d-1\right)^2}[/tex]
Therefore [tex]P=\frac{-297d+378}{\left(d-1\right)^2}+343[/tex].
Hope this helps!
