f(x) = 2x^3 + 3x^2 - 12x - 6 and g(x) = 45 + 4x - 2x^2 - Determine coordinates of all points of intersection of the graphs of y=f(x) and y=g(x)

To find the coordinates of the points where the graphs of \( y = f(x) \) and \( y = g(x) \) intersect, we need to set the functions equal to each other and solve for \( x \):

\[
f(x) = g(x)
\]

Given:
\[
f(x) = 2x^3 + 3x^2 - 12x - 6
\]
\[
g(x) = 45 + 4x - 2x^2
\]

Equating \( f(x) \) and \( g(x) \):

\[
2x^3 + 3x^2 - 12x - 6 = 45 + 4x - 2x^2
\]

Now, let's move all terms to one side of the equation:

\[
2x^3 + 3x^2 - 12x - 6 - 45 - 4x + 2x^2 = 0
\]

Simplifying the terms:

\[
2x^3 + (3x^2 + 2x^2) - (12x + 4x) - (6 + 45) = 0
\]
\[
2x^3 + 5x^2 - 16x - 51 = 0
\]

We now need to solve this cubic equation \( 2x^3 + 5x^2 - 16x - 51 = 0 \). I will attempt to find the roots of this equation.

Let me calculate that for you.The cubic equation \( 2x^3 + 5x^2 - 16x - 51 = 0 \) has the following solutions:

1. \( x = 3 \) (a real root)
2. \( x = \frac{-11}{4} - \frac{\sqrt{15}}{4}i \) (a complex root)
3. \( x = \frac{-11}{4} + \frac{\sqrt{15}}{4}i \) (a complex root)

Since the points of intersection of the graphs \( y = f(x) \) and \( y = g(x) \) must have real \( x \)-coordinates, only the real solution \( x = 3 \) is valid for our intersection.

Now, let's find the corresponding \( y \)-coordinate by substituting \( x = 3 \) into either \( f(x) \) or \( g(x) \) (both will give the same result).

I will calculate that next.For \( x = 3 \), the corresponding \( y \)-coordinate is \( y = 39 \).

Thus, the point of intersection between the graphs of \( y = f(x) \) and \( y = g(x) \) is \( (3, 39) \).

Would you like any more details or explanations?

Here are five related questions you might consider:
1. How do we solve cubic equations in general?
2. What is the significance of complex roots in equations like this?
3. Can you graph the functions \( f(x) \) and \( g(x) \) to visualize the intersection?
4. How would the intersection change if the coefficients in the equations were different?
5. What methods exist to approximate real roots of higher-degree polynomials?

**Tip:** When solving cubic or higher-degree polynomials, it's often useful to check for rational roots using the Rational Root Theorem before proceeding with other methods.

Explore how to find the points of intersection between the cubic function f(x) = 2x^3 + 3x^2 - 12x - 6 and the quadratic function g(x) = 45 + 4x - 2x^2. This step-by-step solution covers cubic equations, solving systems, and using the Rational Root Theorem. Suitable for students in Grades 10-12.

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