Problem 13 - Precalculus

Question

Solve the given system of nonlinear equations. Use a graph to help you avoid any potential extraneous solutions. $$ \left\\{\begin{array}{l} y=x^{3}+8 \\ y=10 x-x^{2} \end{array}\right. $$

Step-by-Step Solution

Answer

Solve \( x^3 + x^2 - 10x + 8 = 0 \) to find intersection points. Each \( x \)-value gives corresponding \( y \)-value.

Step 1: Set Equations Equal

To solve a system of equations, start by setting the two equations equal to each other, since both are equal to \( y \):\[ x^3 + 8 = 10x - x^2 \].

Step 2: Rearrange Into a Single Equation

Bring all terms to one side of the equation to form a single equation: \[ x^3 + x^2 - 10x + 8 = 0 \].

Step 3: Graph the Functions

Plot the functions \( y = x^3 + 8 \) and \( y = 10x - x^2 \) on the same graph. Find the points where they intersect, representing the solutions to the equations.

Step 4: Solve for Real Roots

Use the graph to estimate and verify the roots of the equation from Step 2. The intersections indicate the \( x \)-values that satisfy both original equations.

Step 5: Substitute to Find \( y \)-Values

For each \( x \)-value found in Step 4, substitute back into either \( y = x^3 + 8 \) or \( y = 10x - x^2 \) to find the corresponding \( y \)-value.

Step 6: Verify Solutions

Ensure that each \( (x, y) \) pair satisfies both original equations to confirm there are no extraneous solutions.

Key Concepts

Graphical Solution

Using a graphical solution for nonlinear systems of equations provides a visual representation of where two functions intersect. This approach often helps to avoid mathematically complex errors, such as extraneous solutions. When graphing, you plot both equations on the same Cartesian coordinate plane and visually identify the intersection points, which are the solutions to the system.

In this exercise, by graphing the two equations:

• \( y = x^3 + 8 \)
• \( y = 10x - x^2 \)

You can quickly estimate the solutions where both functions have the same inputs and outputs. It's a reliable way to make sure no potential solutions are overlooked or incorrectly identified.

Graphical solutions are particularly useful in cases where algebraic manipulation may lead to lengthy processes, offering a more intuitive understanding of the behavior of the functions.

Cubic Functions

Cubic functions are a type of polynomial equation characterized by the term \( x^3 \). In the equation from the system \( y = x^3 + 8 \), the cubic function forms an S-shaped curve when graphed.

Some key features of cubic functions include:

• They have three roots, which can be real or complex.
• They demonstrate an inflection point where the curve changes concavity, typically in the center.
• Their slope and curvature constantly change, providing dynamic behaviors across the graph.

When solving systems involving cubic functions, these dynamic characteristics make it crucial to understand how they interact with other types of curves, influencing where and how they intersect with additional functions like quadratics.

Quadratic Functions

Quadratic functions are expressed in the standard form as \( y = ax^2 + bx + c \). For the purpose of this exercise, the quadratic function is given by \( y = 10x - x^2 \). These types of functions create a parabolic shape, which can open either upwards or downwards depending on the coefficient of the \( x^2 \) term.

Important aspects to note:

• The parabola opens downwards since the coefficient of \( x^2 \) is negative.
• The vertex of the parabola represents its highest point, as it opens downwards.
• Quadratics always intersect the y-axis at \( c \), here at the origin essentially, because the equation can be rearranged to intersect at a different constant point.

Quadratic functions' symmetry about their axis also plays an important role in determining solutions when used alongside other functions in a system of equations.

Intersection Points

Intersection points occur where the two curves on a graph meet, representing solutions that satisfy both equations in a system. These points are effectively the 'answers' to the system of equations.

For our system:

• The intersection points are the solutions to the rearranged equation \( x^3 + x^2 - 10x + 8 = 0 \).
• Each respective \( x \)-value found here will cross-check as a solution since it satisfies both equations \( y = x^3 + 8 \) and \( y = 10x - x^2 \).

To find these points, you solve the system by setting the equations equal to each other and then simplify to identify these intersections mathematically and verify through graphical overlap. Checking that these points satisfy both original equations ensures that no extraneous solutions are accepted, keeping the mathematical findings consistent and accurate.