Question

a proton in a magnetic field follows a path on a coordinate grid modeled by the function f(x) = x⁴ - 2x² - 15 what are the x - coordinates of the points on the grid where the proton crosses the x - axis? ○ -√5, √5 ○ 3, -5 ○ √5, √3 ○ -√3, √3

Explanation:

Step1: Set \( f(x) = 0 \)

To find where the proton crosses the \( x \)-axis, we set \( f(x)=0 \), so we have the equation \( x^{4}-2x^{2}-15 = 0 \).

Step2: Substitute \( u = x^{2} \)

Let \( u=x^{2} \), then the equation becomes a quadratic equation in terms of \( u \): \( u^{2}-2u - 15=0 \).

Step3: Solve the quadratic equation

We factor the quadratic equation: \( u^{2}-2u - 15=(u - 5)(u+ 3)=0 \).
Setting each factor equal to zero gives \( u - 5=0 \) or \( u + 3=0 \), so \( u = 5 \) or \( u=-3 \).

Step4: Substitute back \( u=x^{2} \)

Since \( u=x^{2} \), we have \( x^{2}=5 \) or \( x^{2}=-3 \). But \( x^{2}=-3 \) has no real solutions (because the square of a real number cannot be negative). For \( x^{2}=5 \), we take the square root of both sides, so \( x=\pm\sqrt{5} \).

Answer:

\( -\sqrt{5},\sqrt{5} \) (corresponding to the first option: \( -\sqrt{5},\sqrt{5} \))