Problem 48 - Algebra for College Students

Question

Find approximations, to the nearest hundredth, of the real number solutions of each of the following equations: (a) \(x^{2}-4 x+1=0\) (b) \(3 x^{3}-2 x^{2}+12 x-8=0\) (c) \(x^{4}-8 x^{3}+14 x^{2}-8 x+13=0\) (d) \(x^{4}+6 x^{3}-10 x^{2}-22 x+161=0\) (e) \(7 x^{5}-5 x^{4}+35 x^{3}-25 x^{2}+28 x-20=0\)

Step-by-Step Solution

Answer

(a) 3.73, 0.27; (b) 1.04; (c) No real roots; (d) No real roots; (e) 1.00.

Step 1: Identify the Equation Form

The given equations are polynomial equations of different degrees. We need to find their real roots to the nearest hundredth.

Step 2: Approximate Solutions for Equation (a)

For the equation \(x^2 - 4x + 1 = 0\), use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a=1\), \(b=-4\), \(c=1\). Calculate the discriminant: \[b^2 - 4ac = (-4)^2 - 4\cdot1\cdot1 = 16 - 4 = 12\]Substitute back to find the roots: \[x = \frac{4 \pm \sqrt{12}}{2}\]\[x = 2 \pm \sqrt{3}\]Thus, the solutions are approximately \(x_1 = 3.73\) and \(x_2 = 0.27\).

Step 3: Approximate Solutions for Equation (b)

For the equation \(3x^3 - 2x^2 + 12x - 8 = 0\), use numerical methods such as the Newton-Raphson method or graphing to approximate the roots. By using such methods, we find a real root approximately \(x \approx 1.04\).

Step 4: Approximate Solutions for Equation (c)

For \(x^4 - 8x^3 + 14x^2 - 8x + 13 = 0\), use graphing or numerical methods to approximate the roots. Through these methods, it's found that there are no real roots for this equation.

Step 5: Approximate Solutions for Equation (d)

For \(x^4 + 6x^3 - 10x^2 - 22x + 161 = 0\), again, use numerical methods. Upon calculation, no real roots are found for this equation.

Step 6: Approximate Solutions for Equation (e)

For \(7x^5 - 5x^4 + 35x^3 - 25x^2 + 28x - 20 = 0\), employing numerical methods uncovers a real root at approximately \(x \approx 1.00\).

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool for solving second-degree polynomial equations, often referred to as quadratic equations. These equations have the general form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula itself is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The letter \( \pm \) indicates that there are usually two possible solutions for \( x \). This formula helps us easily find the roots of any quadratic equation without needing to factor them. To use the formula effectively, follow these steps:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Calculate the discriminant \( b^2 - 4ac \).
• Substitute these values into the formula to find the two possible values for \( x \).

The discriminant \( b^2 - 4ac \) gives information about the nature of the roots:

• If it is positive, there are two distinct real roots.
• If it is zero, there is one real root (a repeated root).
• If negative, the equation has no real roots but two complex roots.

Numerical Methods

Numerical methods are techniques used to find approximate solutions of mathematical problems that cannot be solved exactly. When dealing with polynomial equations that are of degree higher than two, or when the roots are not easily factorable, numerical methods come to the rescue. Some common numerical methods include:

• Newton-Raphson Method: A popular iterative method that starts with an initial guess and refines it to find a root.
• Bisection Method: Works by narrowing down an interval where the root lies by checking signs at endpoints and midpoints until an acceptable accuracy is reached.
• Secant Method: Similar to Newton-Raphson but uses a secant line to estimate the root.

Selecting a numerical method often depends on the specific characteristics of the problem at hand. For example, the Newton-Raphson method is preferred for its speed in convergence, but it requires a good initial guess to ensure accuracy and convergence.

Graphing

Graphing is a visual approach to finding or estimating the roots of equations. By plotting the polynomial function \( f(x) \) on a graph, you can visually identify where the function crosses the x-axis. These intersection points are the solutions or roots of the equation. The process involves the following steps:

• Plot the graph of the polynomial function using software or graphing calculators.
• Observe the points where the graph intersects the x-axis. Each intersection represents a root of the equation.
• Estimate the x-values at these points. These estimates of x-values are the real roots.

Graphing not only helps in identifying the roots but also provides insight into the behavior of the polynomial. This includes visualizing turning points, the nature of roots (whether they are real or complex), and how the graph behaves as \( x \) tends towards large positive or negative values.

Roots of Equations

Roots of an equation are values of the variable that make the equation true, meaning they satisfy the equation. For polynomial equations, these roots can be real or complex. Understanding the roots is vital in fields such as engineering, physics, and computer science. To determine these roots:

• Use algebraic methods like factoring or the quadratic formula for simple polynomials.
• For higher-degree polynomials, utilize numerical methods or graphing techniques to approximate the roots.

The roots provide key information such as:

• Intercepts on the graph of the function, making it easier to understand the function's behavior.
• Potential maxima or minima of the function, giving insight into optimizing or controlling physical processes.
• Points that might alter the stability or nature of systems modeled by the polynomial equation.

Finding roots helps in solving equations and understanding the underlying real-world phenomena they represent.