Can Someone Please Explain This Tangent/normal Answer Watch

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faa
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#1
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#1
Hi, heres the question:

The curve with equation y=f(x) is such that

dy/dx = 3x^2+4x+k

where k is a constant

Given that C passes through the points (0,-2) and (2,18).

a) Show that k=2 and find an equation for C
b) Show that the line with equation y=x-2 is a tangent to C and find the coordinates of the point of contact.

I can do a)
k=2 c=-2 y=x^3+2x^2+2x-2

but with b, the answer says:

x^3+2x^2+2x-2=x-2
x^3+2x^2+x=0
x(x^2+2x+1)=0
x(x+1)^2=0
REPEATED ROOT THEREFORE TANGENT
point of contact where x=-1
THEREFORE (-1,-3)

can someone please explain where they got those last three lines from.

Cheers guys!
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J.F.N
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(Original post by faa)
but with b, the answer says:

x^3+2x^2+2x-2=x-2
x^3+2x^2+x=0
x(x^2+2x+1)=0
x(x+1)^2=0
REPEATED ROOT THEREFORE TANGENT
point of contact where x=-1
THEREFORE (-1,-3)
Notice that x(x+1)^2=x(x+1)(x+1)=0. So, the only solution is for x=-1, i.e. a repeated root. So x=-1 is plugged into the prior equation to give y=-3.
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john !!
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the repeated root of the equation above is x = -1.
when a curve touches a line, say curve y=f(x) and line y=ax + b
you will have a repeated root for the equation f(x) = ax+b
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faa
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(Original post by J.F.N)
Notice that x(x+1)^2=x(x+1)(x+1)=0. So, the only solution is for x=-1, i.e. a repeated root. So x=-1 is plugged into the prior equation to give y=-3.
why does a repeated root automatically suggest a tangent? Im referring to the suggestion that was said in the mark scheme, not your answer hehe
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faa
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(Original post by mik1a)
the repeated root of the equation above is x = -1.
when a curve touches a line, say curve y=f(x) and line y=ax + b
you will have a repeated root for the equation f(x) = ax+b
(you might have to bear with me in this explanation and use your imagination lol)

is that because the line y=ax+b is acting like the x axis does with a quadratic equation. When you have repeated roots in a quadratic, the x axis acts as a tangent.

Therefore when you say that f(x)=ax+b, you are simply putting the two together, to work this out, but instead of the x-axis as the tangent, you will be using the y=ax+b?
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john !!
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repeated root means it only touches the line once. a quadratic with two distinct roots would cross twice because of its shape...
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faa
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(Original post by mik1a)
repeated root means it only touches the line once. a quadratic with two distinct roots would cross twice because of its shape...
yeh ok ok!! lol, i meant a quadratic that touches it once! lol. Then would my reasoning be correct?
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john !!
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(Original post by faa)
(you might have to bear with me in this explanation and use your imagination lol)

is that because the line y=ax+b is acting like the x axis does with a quadratic equation. When you have repeated roots in a quadratic, the x axis acts as a tangent.

Therefore when you say that f(x)=ax+b, you are simply putting the two together, to work this out, but instead of the x-axis as the tangent, you will be using the y=ax+b?
yes. neat huh?

(Original post by faa)
yeh ok ok!! lol, i meant a quadratic that touches it once! lol. Then would my reasoning be correct?
Sorry I was repliying to what you said eariler.
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