# Where does the line y= 5x-3 meets the curve f(x) = x^2 +3 ?

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Where the line y= 5x-3 meets the curve f(x) = x^2 +3, the x and y-coordinates for both of them are the same. So we can equate the two and solve for x.

5x - 3 = x^2 + 3

=> x^2 - 5x + 6 = 0

=> x^2 - 3x - 2x - 6 = 0

=> x(x - 3) - 2(x - 3) = 0

=> (x - 2 )(x - 3) = 0

=> x = 2 and x = 3

y = 5x - 3 = 7 and y = 12

**The points of the contact are (2 , 7) and (3, 12).**

Given the line y= 5x -3 and the curve f(x) = x^2 +3

We need to find the intersection points between the line and the curve.

==> f(x) = y

==> x^2 + 3 = 5x-3

==> Now we will combine like terms.

==> x^2 - 5x + 6 = 0

Now we will factor.

==> (x-3)(x-2) = 0

==> x 1= 3 ==> y1 = 5x-3 = 5*3-3 = 12

==> x2= 2 ==> y2= 5x-3 = 5*2-3 = 7

**Then the intersection points are ( 2,7) and (3, 12)**