Question

determining solutions of a cubic equation
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: Factor the function

To find the solutions of \( f(x) = x^3 - 5x^2 + 6x \), we first factor the function. Factor out the common factor \( x \):

$$ f(x)=x(x^2 - 5x + 6) $$

Then factor the quadratic expression \( x^2 - 5x + 6 \). We need two numbers that multiply to \( 6 \) and add up to \( - 5 \). The numbers are \( -2 \) and \( -3 \), so:

$$ x^2 - 5x + 6=(x - 2)(x - 3) $$

Thus, the factored form of the function is:

$$ f(x)=x(x - 2)(x - 3) $$

Step2: Find the roots

To find the solutions (roots) of the equation \( f(x)=0 \), we set each factor equal to zero:

• For \( x = 0 \): \( x=0 \) is a solution.
• For \( x - 2=0 \): Solving for \( x \), we get \( x = 2 \).
• For \( x - 3=0 \): Solving for \( x \), we get \( x = 3 \).

If we were to graph the function \( f(x)=x^3 - 5x^2 + 6x \), the \( x \)-intercepts (where the graph crosses the \( x \)-axis, which correspond to the solutions of \( f(x) = 0 \)) would be at \( x = 0 \), \( x = 2 \), and \( x = 3 \).

Answer:

The solutions to the equation \( f(x)=x^3 - 5x^2 + 6x = 0 \) are \( x = 0 \), \( x = 2 \), and \( x = 3 \). So we would check the boxes corresponding to \( 0 \), \( 2 \), and \( 3 \).