Find The Gradient of the tangent to the Curve y=x²-3x at the point (2,-2)

A non-calculus solution follows afterward.

For a calculus solution, the gradient is the first derivative with the point put in.

y = -x²+3x

dy/dx or y’ = -2x +3 is the gradient at any point.

but the tangent is at x = 2 so put it in

y’ = -(2*2)+3 = -1 is the gradient of the tangent at point (2,2)

Non-calculus solution:

y = -x²+3x 1️⃣but slope of tangent m = (y-2)/(x-2)2️⃣

put 1️⃣ into 2️⃣ and rearrange ⟹ mx-2m+x²-3x+2 = 0

or x²+(m-3)x+2–2m =0

but for a tangent the discriminant b²-ac = 0

(m-3)²-4*1*(2–2m) = 0

m²+2m+1 = 0

(m+1)(m+1) = 0

m = -1 is the is the gradient of the tangent at point (2,2)

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