Problem 48 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ x^{2}+4 x-\frac{7}{x^{2}+4 x+5}=1 $$

Step-by-Step Solution

Answer

The given equation is: \( x^2 + 4x - \frac{7}{x^2 + 4x + 5} = 1 \) After simplifying, finding a common denominator, and expanding, we get a fourth-degree polynomial equation: \( x^4 + 8x^3 + 20x^2 + 16x - 13 = 0 \) Since this is difficult to solve algebraically, we can use numerical methods or calculators to approximate the solutions. Once found, substitute the solutions back into the original equation to verify their validity.

Step 1: Simplify the equation

To simplify the equation, we need to subtract 1 from both sides to bring everything to one side of the equation. This gives us: \( x^2 + 4x - \frac{7}{x^2 + 4x + 5} - 1 = 0 \)

Step 2: Combine like terms

We need to combine the terms involving x squared and the constant terms. This gives us: \( x^2 + 4x - 1 - \frac{7}{x^2 + 4x + 5} = 0 \)

Step 3: Find a common denominator

In order to eliminate the fraction, we'll multiply both sides of the equation by the denominator (\(x^2 + 4x + 5\)), which yields: \( (x^2 + 4x + 5)(x^2 + 4x - 1) - 7 = 0 \)

Step 4: Expand and simplify the equation

Now, we'll expand the left side of the equation and combine like terms: \( (x^2 + 4x - 1)(x^2 + 4x + 5) - 7 = \\ x^4 + 4x^3 + 5x^2 + 4x^3 + 16x^2 + 20x - x^2 - 4x - 5 - 7 = \\ x^4 + 8x^3 + 20x^2 + 16x - 13 = 0 \)

Step 5: Solve the quadratic equation

At this point, we have a fourth-degree polynomial equation, which is difficult to solve algebraically. We can, however, use various numerical methods like Newton's Method or use calculators to approximate the solutions. Alternatively, you can also ask your teacher if this was the intended difficulty level of the problem, and check if there were any algebraic errors in the process or if there's an easier way to solve this particular equation.

Step 6: Check the solution

Once you have the approximate solutions, substitute them back into the original equation to make sure they satisfy the equation. If any solutions don't satisfy the equation, then they are extraneous or were introduced through the approximation process.

Key Concepts

Solving Polynomial Equations

Working with polynomial equations is a foundational concept in algebra and pre-calculus. A polynomial equation features variables raised to whole number exponents. These can range from simple linear equations to complex higher-degree expressions.

In the case of our exercise, moving from a quadratic equation to a fourth-degree polynomial presented an unexpected complexity. To solve a polynomial equation, one typically starts by arranging the equation in standard form, with all terms on one side and the constant on the other. That was our first step. Simplification often follows, consolidating like terms as seen in step 2.

Finding roots for higher-degree polynomials can be challenging. While quadratic equations can be solved using the quadratic formula, higher-degree polynomials may require factoring, synthetic division, or graphing to find zeros. If these algebraic methods are not applicable or too complex, we turn to numerical methods, which give us an approximate solution. Students can then use these approximate values to verify the accuracy of the solution, ensuring they satisfy the original equation.

Importance of Zeroes in Polynomials

Zeroes, or roots, are the values of the variable that satisfy the polynomial equation when set to zero. They play a crucial role in graphing polynomials and understanding their behavior. Finding these roots can sometimes involve using the Rational Root Theorem or Descartes' Rule of Signs to simplify the process.

Unforeseen Challenges

Solving polynomial equations like the one presented here could involve unexpected steps. If the degree of the polynomial creates substantial difficulty, it might be worth revisiting the initial problem to ensure no simplification was overlooked. This vigilance can prevent unnecessary complexity in solving what might be a basic quadratic equation.

Common Denominators in Algebra

Dealing with fractions in algebra often requires finding common denominators. This is a critical step in simplification, enabling the combination of fractions or the elimination of a fraction altogether.

In our exercise, we approach step 3 by looking for a common denominator to combine the terms of the equation effectively. The common denominator is found by taking the non-fractional part of the equation and using it to eliminate the fractional component. This process leads to a single polynomial expression without fractions, simplifying further manipulation.

When to Find a Common Denominator

You should find a common denominator whenever you are adding, subtracting, or comparing fractions. The least common denominator (LCD) is the smallest number that can be divided evenly by all denominators in the problem. By converting all terms to have the LCD, you ensure accuracy in combining terms and streamline the overall solving process.

Impact on Simplifying Equations

A common denominator can dramatically simplify the complexity of an equation. Without this step, solving an equation with multiple fractional terms can become arduous and prone to error. When students find the common denominator, it's like finding a 'translator' that allows each term to speak the same 'fractional language,' leading to a much smoother problem-solving journey.

Numerical Methods in Calculus

Calculating exact solutions for complex equations, especially those of higher degree polynomials or transcendental functions, often goes beyond standard algebraic methods. That's where numerical methods come in, a suite of techniques in calculus designed for approximating solutions to otherwise intractable problems.

In step 5 of our exercise, such methods are suggested because the fourth-degree polynomial is not easily solvable using algebraic methods. Techniques like Newton's Method, also known as the Newton-Raphson method, use iterations to close in on a root of the equation. This method starts with a guessed value and refines it incrementally to converge to a solution.

Choosing an Appropriate Method

Depending on the nature of the problem, different numerical methods may be more suitable. For instance, bisection or secant methods are also employed to find roots, while series expansion and differential equations might be approached through Euler's Method or the Runge-Kutta methods.

Role in Analysis and Engineering

Numerical methods are particularly important in real-world applications where exact solutions are not practical. These techniques are used extensively in engineering, physics, and financial models to make predictions and solve complex differential equations. These methods generally involve iterative calculations, which can be executed efficiently using computational tools. The trade-off of using numerical methods is precision for practicality, as we accept approximations that are close enough to the true solution for practical purposes.