Find all the x-intercepts of the curve y=4x^4-17x^2-15
4x^4-17x^2-15 = (4x^2+3)(x^2-5)
so, it should be easy now ...
To find the x-intercepts of a curve, we need to determine the values of x when y equals zero. In other words, we are looking for the x-coordinates of the points where the curve intersects the x-axis.
In this case, we are given the equation of the curve y = 4x^4 - 17x^2 - 15.
To find the x-intercepts, we set y equal to zero and solve for x:
0 = 4x^4 - 17x^2 - 15
Now, this equation is a quadratic equation in terms of x^2. To solve it, we can use factoring or the quadratic formula. Let's use factoring in this case.
First, let's rewrite the equation in terms of x^2:
0 = (2x^2 - 5)(2x^2 + 3)
To find the x-intercepts, we can set each factor equal to zero:
2x^2 - 5 = 0
and
2x^2 + 3 = 0
For the first equation, we have:
2x^2 - 5 = 0
Adding 5 to both sides, we get:
2x^2 = 5
Dividing both sides by 2, we have:
x^2 = 5/2
Taking the square root of both sides, we obtain:
x = ±√(5/2)
For the second equation, we have:
2x^2 + 3 = 0
Subtracting 3 from both sides, we get:
2x^2 = -3
Dividing both sides by 2, we have:
x^2 = -3/2
Since the square root of a negative number is not a real number, this equation has no real solutions.
Therefore, the x-intercepts of the curve y = 4x^4 - 17x^2 - 15 are:
x = √(5/2) and x = -√(5/2).