Question 881469
{{{y=mx+3}}}
{{{y=x^2+c}}}
Set them equal to each other.
{{{mx+3=x^2+c}}}
{{{x^2-mx+c-3=0}}}
Since they only intersect at one point, the discriminant equals zero.
{{{m^2-4(c-3)=0}}}
{{{m^2=4(c-3)}}}
If order for {{{m}}} to be real, {{{c-3>=0}}}, or {{{c>=3}}}
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When {{{c=3}}},{{{m=0}}}
Then the line becomes, {{{y=3}}} and the parabola becomes, {{{y=x^2+3}}}
{{{y=3}}}
{{{x^2+3=3}}}
{{{x=0}}}
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{{{drawing(300,300,-5,5,-1,9,grid(1),circle(0,3,0.2),blue(line(-10,3,10,3)),graph(300,300,-5,5,-1,9,3,x^2+3))}}}

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When {{{c=4}}},{{{m=2}}}
Then the line becomes, {{{y=x+3}}} and the parabola becomes, {{{y=x^2+4}}}
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{{{drawing(300,300,-5,5,-1,9,circle(1,5,0.2),grid(1),graph(300,300,-5,5,-1,9,2x+3,x^2+4))}}}
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{{{2x+3=x^2+4}}}
{{{x^2-2x+1=0}}}
{{{(x-1)^2=0}}}
{{{x=1}}}
{{{y=2(1)+3=5}}}