2. REASON QUANTITATIVELY Explain why the \( x \)-coordinates of the points of intersection of \( f(x)=x^{3}+x^{2}-14 x-4 \) and \( g(x)=x^{3}-3 x^{2}-6 x+28 \) represent the solutions to \( x^{3}+x^{2}-14 x-4=x^{3}-3 x^{2}-6 x+28 \). Then use the graphs to solve the equation.

Certainly! Let's break down the problem step by step to understand why the \( x \)-coordinates of the intersection points of the functions \( f(x) \) and \( g(x) \) represent the solutions to the equation \( f(x) = g(x) \), and then solve the equation using this reasoning. ## Understanding the Relationship Between Intersection Points and Solutions ### Quantitative Reasoning 1. **Intersection Points Definition**: - The **points of intersection** between two graphs \( f(x) \) and \( g(x) \) are the points where both functions have the same \( y \)-value for the same \( x \)-value. In other words, at these points, \( f(x) = g(x) \). 2. **Setting Up the Equation**: - To find the \( x \)-coordinates where \( f(x) \) and \( g(x) \) intersect, we set the two functions equal to each other: \[ f(x) = g(x) \] 3. **Solving for \( x \)**: - By solving the equation \( f(x) = g(x) \), we determine the specific \( x \)-values where the two functions yield the same output. These \( x \)-values correspond to the horizontal positions of the intersection points on the graph. Thus, **solving \( f(x) = g(x) \)** directly gives us the \( x \)-coordinates of the points where the graphs of \( f(x) \) and \( g(x) \) intersect. ## Solving the Equation Using Graphical Reasoning Given the functions: \[ f(x) = x^{3} + x^{2} - 14x - 4 \] \[ g(x) = x^{3} - 3x^{2} - 6x + 28 \] ### Step 1: Set \( f(x) \) Equal to \( g(x) \) \[ x^{3} + x^{2} - 14x - 4 = x^{3} - 3x^{2} - 6x + 28 \] ### Step 2: Simplify the Equation Subtract \( x^{3} \) from both sides to eliminate the cubic terms: \[ x^{2} - 14x - 4 = -3x^{2} - 6x + 28 \] ### Step 3: Bring All Terms to One Side Add \( 3x^{2} + 6x - 28 \) to both sides: \[ x^{2} + 3x^{2} - 14x + 6x - 4 - 28 = 0 \] \[ 4x^{2} - 8x - 32 = 0 \] ### Step 4: Simplify the Quadratic Equation Divide the entire equation by 4 to make it simpler: \[ x^{2} - 2x - 8 = 0 \] ### Step 5: Solve the Quadratic Equation Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] For the equation \( x^{2} - 2x - 8 = 0 \): - \( a = 1 \) - \( b = -2 \) - \( c = -8 \) Plugging in the values: \[ x = \frac{-(-2) \pm \sqrt{(-2)^{2} - 4(1)(-8)}}{2(1)} \] \[ x = \frac{2 \pm \sqrt{4 + 32}}{2} \] \[ x = \frac{2 \pm \sqrt{36}}{2} \] \[ x = \frac{2 \pm 6}{2} \] This gives two solutions: \[ x = \frac{2 + 6}{2} = \frac{8}{2} = 4 \] \[ x = \frac{2 - 6}{2} = \frac{-4}{2} = -2 \] ### Step 6: Interpret the Solutions The solutions \( x = 4 \) and \( x = -2 \) are the \( x \)-coordinates where the graphs of \( f(x) \) and \( g(x) \) intersect. ### Graphical Confirmation (Optional) If you were to graph both functions: - **At \( x = 4 \)**: \[ f(4) = 4^{3} + 4^{2} - 14(4) - 4 = 64 + 16 - 56 - 4 = 20 \] \[ g(4) = 4^{3} - 3(4)^{2} - 6(4) + 28 = 64 - 48 - 24 + 28 = 20 \] - **At \( x = -2 \)**: \[ f(-2) = (-2)^{3} + (-2)^{2} - 14(-2) - 4 = -8 + 4 + 28 - 4 = 20 \] \[ g(-2) = (-2)^{3} - 3(-2)^{2} - 6(-2) + 28 = -8 - 12 + 12 + 28 = 20 \] Both functions yield the same \( y \)-value at \( x = 4 \) and \( x = -2 \), confirming these points as intersections. ## **Final Answer** The solutions to the equation \( f(x) = g(x) \) are: \[ \boxed{\,x = -2\quad\text{and}\quad x = 4\,} \]