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Featured 2 months ago

From

and

we get

now squaring both sides

now subtracting side by side

Featured 3 weeks ago

#"the equation of a line in "color(blue)"point-slope form"# is.

#â€¢color(white)(x)y-y_1=m(x-x_1)#

#"where m is the slope and "(x_1,y_1)" a point on the line"#

#"we require to find "m" and "(x_1,y_1)#

#"the equation of a line in "color(blue)"slope-intercept form"# is.

#â€¢color(white)(x)y=mx+b#

#"rearrange "2x+3y=12" into this form"#

#rArr3y=-2x+12rArry=-2/3x+4#

#rArrm=-2/3#

#"to find a point on the line choose any value for x "#

#"substitute into the equation and evaluate for y"#

#x=1toy=-2/3+4=10/3#

#"using "m=-2/3" and "(x_1,y_1)=(1,10/3)#

#y-10/3=-2/3(x-1)larrcolor(blue)"in point-slope form"#

Featured 1 week ago

#"using the "color(blue)"rules of radicals"#

#â€¢color(white)(x)sqrt(ab)hArrsqrtaxxsqrtb#

#â€¢color(white)(x)(sqrta+sqrtb)(sqrta-sqrtb)=a^2-b^2#

#"let's begin by simplifying the given radicals"#

#sqrt18=sqrt(9xx2)=sqrt9xxsqrt2=3sqrt2#

#sqrt12=sqrt(4xx3)=sqrt4xxsqrt3=2sqrt3#

#sqrt8=sqrt(4xx2)=sqrt4xxsqrt2=2sqrt2#

#sqrt48=sqrt(16xx3)=sqrt16xxsqrt3=4sqrt3#

#rArr(sqrt18+sqrt12)/(sqrt8-sqrt48)=(3sqrt2+2sqrt3)/(2sqrt2-4sqrt3)#

#"we now require to "color(blue)"rationalise the denominator"#

#"that is, eliminate the radicals from the denominator"#

#"multiply numerator/denominator by the "color(blue)"conjugate"#

#"of the denominator"#

#"the conjugate of "2sqrt2-4sqrt3 " is "2sqrt2color(red)(+)4sqrt3#

#=((3sqrt2+2sqrt3)(2sqrt2+4sqrt3))/((2sqrt2-4sqrt3)(2sqrt2+4sqrt3))#

#"expand the factors using FOIL gives"#

#=(12+12sqrt6+4sqrt6+24)/(8-48)#

#=(36+16sqrt6)/(-40)#

#=36/(-40)+(16sqrt6)/(-40)=-9/10-2/5sqrt6#

Featured 1 week ago

Proof:-#" "Auu(BnnC)=(AuuB)nn(AuuC)# Let,

#" "x in Auu(BnnC)#

#=>x in A vv x in (BnnC)#

#=>x in A vv (x in B ^^ x in C)#

#=>(x in A vv x in B) ^^ (x in A vv x in C)#

#=>x in (A uu B) ^^ x in (A uu C)#

#=>x in (AuuB)nn(AuuC)#

#x in Auu(BnnC)=>x in (AuuB)nn(AuuC)#

#=>color(red)(Auu(BnnC)sube(AuuB)nn(AuuC)# Let,

#" "y in (AuuB)nn(AuuC)#

#=>y in (A uu B) ^^ y in (A uu C)#

#=>(y in A vv y in B) ^^ (y in A vv y in C)#

#=>y in A vv (y in B ^^ y in C)#

#=>y in A vv y in (BnnC)#

#=>y in Auu(BnnC)#

#x in (AuuB)nn(AuuC)=>x in Auu(BnnC)#

#=>color(red)((AuuB)nn(AuuC)subeAuu(BnnC)#

From the both red part , we get by using the rule of equal set,

#color(red)(ul(bar(|color(green)(Auu(BnnC)=(AuuB)nn(AuuC)))|# Hope it helps...

Thank you...

Featured 5 days ago

We can easily rewrite

Subtracting

which means that

'

i.e.

Now, for a polynomial to vanish, one of its factors have to be zero. So, either

If

or if

or, finally,

Featured 5 days ago

The distance is

The easiest way is to use the distance formula, which is kinda tricky:

That looks really complex, but if you take it slowly, I'll try and help you through it.

So let's call

Let's call

Let's plug these numbers into the distance formula:

This is quite the hard subject, and is best taught by someone who knows how to explain well! This is a really good video about the distance formula:

Khan Academy distance formula video

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