Find the coordinates of all points where the curve y=x³+2x-15x cuts the x-axis
To find the coordinates of the points where the curve intersects the x-axis, we need to set y equal to zero (since the y-coordinate of any point on the x-axis is always zero) and solve for x.
Setting y = 0 in the equation y = x³ + 2x - 15x, we have:
0 = x³ + 2x - 15x
Simplifying this expression, we get:
0 = x³ - 13x
Factor out an x:
0 = x(x² - 13)
Setting each factor equal to zero:
x = 0
x² - 13 = 0
For x = 0, the y-coordinate is also 0, so the first point where the curve intersects the x-axis is (0, 0).
For x² - 13 = 0, we can solve this equation for x by taking the square root of both sides:
x² = 13
x = ±√13
So, the curve intersects the x-axis at the points (√13, 0) and (-√13, 0).
To find the coordinates of the points where the curve intersects the x-axis, we set y to zero and solve for x.
Setting y = 0, we have:
0 = x³ + 2x - 15x
Combining like terms, we get:
0 = x³ - 13x
Factoring out x, we have:
0 = x(x² - 13)
Setting each factor equal to zero, we have:
x = 0
and
x² - 13 = 0
Solving for x² - 13 = 0, we get:
x² = 13
Taking the square root of both sides, we find:
x = √13 and x = -√13
Therefore, the x-coordinates of the points where the curve intersects the x-axis are:
x = 0, √13, and -√13.
Since the y-coordinate of a point on the x-axis is always zero, the coordinates of the points are:
(0, 0), (√13, 0), and (-√13, 0).