Question

graph the function f(x) = x³ - 5x² + 6x using the graphing calculator. what are the solutions to the related equation? check all that apply. -3 -2 -1 0 1 2 3

Explanation:

Step1: Set the function to zero

To find the solutions of the related equation, we set \( f(x) = 0 \), so we have the equation \( x^{3}-5x^{2}+6x = 0 \).

Step2: Factor out the common term

Factor out an \( x \) from the left - hand side of the equation: \( x(x^{2}-5x + 6)=0 \).

Step3: Factor the quadratic expression

Factor the quadratic \( x^{2}-5x + 6 \). We need two numbers that multiply to \( 6 \) and add up to \( - 5 \). The numbers are \( -2 \) and \( -3 \), so \( x^{2}-5x + 6=(x - 2)(x - 3) \). Then the equation becomes \( x(x - 2)(x - 3)=0 \).

Step4: Solve for x

Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So we set each factor equal to zero:

• If \( x=0 \), the equation is satisfied.
• If \( x - 2=0 \), then \( x = 2 \).
• If \( x - 3=0 \), then \( x = 3 \).

Answer:

0, 2, 3 (corresponding to the options: 0, 2, 3)