How many real solutions (intersections) will there be in this system of two equations?

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Answer 1 · 2018-04-24T04:53:06

Two solutions - #((-1+sqrt17)/2,(1+sqrt17)/2)# and #((-1-sqrt17)/2,(1-sqrt17)/2)#

As #y=x^2-3# and #y=x+1#

we have #x^2-3=x+1# or #x^2-x-4=0#

Now as discriminant of quadratic equation is #(-1)^2-4*1*(-4)=17# a positive real number

we have two real solutions. These are given by

#x=((-1)+-sqrt((-1)^2-4*1*(-4)))/2=(-1+-sqrt17)/2#

i.e. #x=(-1+sqrt17)/2# and #(-1-sqrt17)/2#

and as #y=x+1#, #y=(1+sqrt17)/2# and #(1-sqrt17)/2#

and solutions are #((-1+sqrt17)/2,(1+sqrt17)/2)# and #((-1-sqrt17)/2,(1-sqrt17)/2)#

graph{(y-x-1)(y-x^2+3)=0 [-10, 10, -5, 5]}