Problem 25 - A Graphical Approach to College Algebra

Question

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &-2 x^{3}-x^{2}+3 x=0\\\ &[-4,4] \text { by }[-10,10] \end{aligned}$$

Step-by-Step Solution

Answer

The complex solutions are \(x = 0, x = 1, x = \frac{3}{2}\).

Step 1: Factor out the common term

The given equation is \(-2x^3 - x^2 + 3x = 0\).Notice that each term in the polynomial shares a common factor of \(x\).Therefore, factor \(x\) out of the polynomial:\[ x(-2x^2 - x + 3) = 0 \]

Step 2: Solve for the first solution

From the factored equation, we first set \(x = 0\):This gives a solution:\[ x = 0 \]

Step 3: Factor the quadratic expression

Now, solve \(-2x^2 - x + 3 = 0\).To factor:First, multiply the coefficient of \(x^2\) i.e., -2, with the constant term 3, which gives -6.We need two numbers that multiply to -6 and add to -1 (coefficient of \(x\)). These numbers are +2 and -3.

Step 4: Split the middle term and factor by grouping

Using the numbers found, rewrite the quadratic:\[ -2x^2 + 2x - 3x + 3 = 0 \]Group the terms:\[ (-2x^2 + 2x) + (-3x + 3) = 0 \]Factor each group:\[ 2x(-x + 1) - 3(x - 1) = 0 \]Notice we can factor out a common binomial factor \((-x + 1)\):\[ (-x + 1)(2x - 3) = 0 \]

Step 5: Solve the factored quadratic equation

Set each factor equal to zero:For \(-x + 1 = 0 \): \[ x = 1 \]For \(2x - 3 = 0 \):\[ 2x = 3 \]Divide both sides by 2:\[ x = \frac{3}{2} \]

Step 6: Compile all solutions

The solutions for the original equation \(-2x^3 - x^2 + 3x = 0\) are:\[ x = 0, x = 1, x = \frac{3}{2} \]

Step 7: Verify and support real solutions using graphing

Graph \(y_1 = -2x^3 - x^2 + 3x\) in the window \([-4, 4]\) for x and \([-10, 10]\) for y.Use the graphing calculator to verify the solutions:Look where the graph intersects the x-axis, which confirms the real solutions.The graph will cross the x-axis at \(x = 0\), \(x = 1\), and \(x = \frac{3}{2}\).

Key Concepts

Factoring polynomials

Factoring polynomials involves breaking down a complex equation into simpler components. This process is mostly about identifying and extracting common factors from polynomial terms to simplify equations. For the equation \[-2x^3 - x^2 + 3x = 0\], the first step is to find a common factor in all terms. Here, all terms share an \(x\) factor.

• Identify common factors: Always look for numbers, variables, or expressions that are shared among all terms in the polynomial. Pull these out to simplify.


• Write the expression in terms of factored components: This often reduces the complexity of the equation and sets the stage for easier solutions.


• Solve simpler equations: Once factored, each component can be set to zero to find solutions, as with \(x(-2x^2 - x + 3) = 0\).

By factoring \(x\), the polynomial turns into simpler expressions leading to straightforward solutions such as \(x = 0\). This technique is vital because it reduces a complex problem into a manageable one, often providing immediate results without complex calculations.

Graphical analysis

Graphical analysis offers a visual perspective for understanding functions and their solutions. When we graph the polynomial \(-2x^3 - x^2 + 3x = y_1\) in the viewing window \([-4, 4]\) for \(x\) and \([-10, 10]\) for \(y\), we can visually confirm the solutions.

• Identifying intercepts: The points where the graph crosses the x-axis are solutions of the equation, specifically \(x = 0, 1, \frac{3}{2}\). This provides an excellent way to confirm solutions derived algebraically.


• Analyzing graph behavior: The curve's shape provides insights about the function's degree and leading coefficients. Higher degrees often mean more twists and turns.


• Using technology: Graphing calculators or software like Desmos make it easy to check real roots and behavior within specific windows.

Graphical analysis is essential because it allows you to verify algebraic solutions, see exact solution points, and understand the structure of the polynomial quickly.

Quadratic equations

Quadratic equations are a specific type of polynomial characterized by the general form \(ax^2 + bx + c = 0\). In our problem, after factoring out the \(x\), we solve \(-2x^2 - x + 3 = 0\), which is a quadratic equation.

• Recognizing quadratics: These equations include terms up to \(x^2\) and often appear as part of larger polynomial expressions.


• Factoring method: One efficient way to solve these is factoring, as shown with \(-2x^2 + 2x - 3x + 3\).


• Using solutions: Often quadratics can be broken down into simpler binomial expressions such as \((-x + 1)(2x - 3) = 0\), which was done by grouping and identifying suitable pairs that sum to \(b\) and product \(ac\).


• Checking solutions: Verify results by substituting back into the original equation or by confirming intercepts using graphical analysis.

Understanding quadratic equations is crucial because they frequently arise in algebra and are essential for unlocking solutions across various mathematical contexts. Their solutions, or roots, help us understand the function's behavior and form a foundational tool for solving higher-degree polynomials.