Find any intercepts.?

Question

y^2=x^3-4x

Answer 1

Given the curve

y^2 = x^3 - 4x

If x = 0, y^2 = 0

This equation admits 2 equal solutions y1 = y2 = 0.

Hence O(0,0) is an intercept where y-axis is tangent to the curve.

Other x-intercepts are points with x-coordinates satisfying y = 0, or

x^3 - 4x = 0

x(x^2 -4) = 0

x(x + 2)(x - 2) = 0

Other than the solution x1 = 0 already mentioned, other solutions to the equation above are x2 = -2 and x3 = +2

There are three intercepts:

(-2,0), (0,0), (+2,0)

Answer 2

Intercepts are where x = 0 or y = 0.

Solving for x = 0

y^2 = 0^3 - 4*0

y^2 = 0

y = 0

so, (0,0) is an intercept.

Now, solve for y = 0

0^2 = x^3 - 4x

0 = x^3 - 4x

factor

0 = x(x^2 - 4)

note that x^2 - 4 is the difference of squares (you should learn to see these, and the formula for factoring  (a^2 - b^2 = (a-b)(a+b) )

so...

0 = x(x-2)(x+2)

by the zero property, one of the factors must be 0 so...

x = 0, x = 2 or x = -2

the intercepts are

(0,0), (2,0) and (-2,0)