Question

1. physics a proton in a magnetic field follows a path on a coordinate grid modeled by the function f(x)=x^4 - 2x^2 - 15. what are the x - coordinates of the points on the grid where the proton crosses the x - axis?

Explanation:

Step1: Set the function equal to 0

Set $f(x)=0$, so $x^{4}-2x^{2}-15 = 0$. Let $u = x^{2}$, then the equation becomes $u^{2}-2u - 15=0$.

Step2: Factor the quadratic equation

Factor $u^{2}-2u - 15$ as $(u - 5)(u+ 3)=0$.

Step3: Solve for u

Using the zero - product property, we have $u - 5=0$ or $u + 3=0$. So $u=5$ or $u=-3$.

Step4: Substitute back $u = x^{2}$

Since $u = x^{2}$, when $u = 5$, we have $x^{2}=5$, then $x=\pm\sqrt{5}$. When $u=-3$, $x^{2}=-3$ has no real solutions because the square of a real number cannot be negative.

Answer:

$x = \sqrt{5},x=-\sqrt{5}$