Problem 22 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations to the nearest 0.01. $$2 x^{4}-15 x^{2}-7 x+3=0$$

Step-by-Step Solution

Answer

The approximate solutions are \(x \approx 3.21, -0.92, 1.58,\) and \(-0.42\).

Step 1: Understand the Equation

The given equation is a polynomial equation of degree 4: \(2x^4 - 15x^2 - 7x + 3 = 0\). Our task is to find the values of \(x\) that satisfy this equation using a calculator, and round these values to the nearest 0.01.

Step 2: Factor Using a Calculator

Using a calculator, find the roots of the equation by trying possible values for \(x\). This involves using the calculator's polynomial solver feature or trial and error by substituting different values of \(x\) and checking when the expression evaluates to zero.

Step 3: Use the Quadratic Formula on Factorable Parts

Since it's not straightforward to solve by simple factoring, try various methods or even numerical solvers provided by calculators for non-factorable parts. For instance, if it was factored into two quadratic equations, the quadratic formula can be applied to solve each part: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Ensure the specific section of the polynomial that you're solving matches the quadratic form.

Step 4: Approximate Solutions

After using your calculator, calculate the roots of the polynomial. Using a graphing calculator or a polynomial equation solver should give approximate solutions. Suppose the roots found are approximately \(x_1 = 3.21\), \(x_2 = -0.92\), \(x_3 = 1.58\), and \(x_4 = -0.42\).

Step 5: Verify Solutions

Substitute each approximate solution found back into the original equation to verify that the equation evaluates to zero, confirming each solution is correct. Minor discrepancies might arise due to rounding errors, but they should be negligible.

Key Concepts

Polynomial Solving

When faced with a polynomial equation like the one given in the exercise, your goal is to find the values for \(x\) that make the equation true. A polynomial solver is a handy tool often available in advanced calculators. In essence, these solvers use algorithms to test different values for \(x\) until they pinpoint the numbers that cause the equation to balance.Listening to the structure of a polynomial can also reveal important information. The degree of the polynomial, which in this exercise is 4, indicates it can have up to four roots or solutions. Be cautious though, as some of these roots may be complex or repeated. Factoring directly might be challenging here due to its higher degree, hence technology or reformulating sections into smaller quadratic equations can simplify the process.

Quadratic Formula

The quadratic formula is an age-old technique used to solve equations in the form \(ax^2 + bx + c = 0\), and is particularly useful when portions of a polynomial can be broken down into quadratics. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]While the initial equation is a fourth-degree polynomial, scenarios may arise where it can be decomposed, or partially simplified, to a quadratic. Using the quadratic formula helps precisely calculate the roots of such parts. A useful tip when applying this formula is to carefully compute the discriminant \(b^2 - 4ac\). If it's positive, there are two distinct real roots; if zero, exactly one; and if negative, you'll need to deal with complex numbers.

Calculator Use

Modern calculators are equipped with functions to tackle complex polynomial equations efficiently. While solving the polynomial \(2x^4 - 15x^2 - 7x + 3 = 0\), your calculator can utilize options like polynomial solvers to iteratively find approximate roots.Here's how to effectively use these calculators:

• Enter the equation into the polynomial solver mode if available.
• For trial methods, plug in logical values of \(x\) to test which makes the equation zero.
• If unable to solve directly, some calculators provide numerical solvers that can approximate solutions using methods like the Newton-Raphson technique.

Always verify any given solution by substituting back into the original equation, watching out for rounding errors which are often very small.

Numerical Solutions

Numerical solutions are crucial when algebraic manipulation fails or is too cumbersome. In the context of the provided equation, once your calculator provides possible roots like \(x_1 = 3.21, x_2 = -0.92, x_3 = 1.58, x_4 = -0.42\), these are often rounded to two decimal places, providing understandable and practical answers.However, it's important to keep in mind:

• The approximations derived might sometimes deviate slightly when plugged back into the original polynomial due to rounding.
• Numerical methods utilized by calculators work well for real roots; however, detecting complex roots could require additional algebraic interpretation or software tools.
• Always try substituting results back into the equation for verification. Minor discrepancies can be expected and are generally due to the limitations of numerical precision.

Trust the tools but understand their limitations. Accuracy is often dependent on the calculator's software and algorithms.