Which Of The Following Equations Have Exactly One Solution

Equations are the cornerstone of mathematics, serving as powerful tools to model and solve a myriad of problems across various disciplines. Determining the number of solutions an equation possesses is a fundamental aspect of mathematical analysis. An equation with exactly one solution signifies a unique value that satisfies the equation, making it a critical concept in algebra and beyond.

Identifying Equations with Exactly One Solution

The quest to find equations with exactly one solution involves understanding the nature of different types of equations and the methods used to solve them. Here, we delve into various equations and explore how to determine if they have a unique solution.

Linear Equations

A linear equation is an equation of the form ax + b = 0, where a and b are constants, and x is the variable. The hallmark of a linear equation is that the variable is raised to the first power.

• Solving Linear Equations: Linear equations are typically solved by isolating the variable on one side of the equation.

  • Example: 2x + 3 = 7

    • Subtract 3 from both sides: 2x = 4
    • Divide both sides by 2: x = 2

• Unique Solution: A linear equation ax + b = 0 has exactly one solution if a ≠ 0. In this case, the unique solution is x = -b/a. If a = 0 and b = 0, the equation becomes 0x + 0 = 0, which is true for all x, meaning there are infinitely many solutions. If a = 0 and b ≠ 0, the equation becomes 0x + b = 0, which is never true, meaning there are no solutions.

Quadratic Equations

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The highest power of the variable is two.

• Solving Quadratic Equations: Quadratic equations can be solved using several methods:

  • Factoring: Expressing the quadratic expression as a product of two linear factors.
  • Completing the Square: Transforming the equation into the form (x - h)² = k.
  • Quadratic Formula: Using the formula x = (-b ± √(b² - 4ac)) / (2a) to find the solutions.

• Discriminant: The discriminant, Δ = b² - 4ac, determines the nature of the solutions:

  • If Δ > 0, there are two distinct real solutions.
  • If Δ = 0, there is exactly one real solution (a repeated root).
  • If Δ < 0, there are no real solutions (two complex solutions).

• Unique Solution: A quadratic equation has exactly one solution when Δ = 0. In this case, the solution is x = -b / (2a).

Cubic Equations

A cubic equation is an equation of the form ax³ + bx² + cx + d = 0, where a, b, c, and d are constants, and a ≠ 0. The highest power of the variable is three.

• Solving Cubic Equations: Solving cubic equations can be more complex than solving linear or quadratic equations. Methods include:

  • Factoring: Factoring the cubic expression into linear and quadratic factors.
  • Cardano's Method: A formula for finding the roots of a cubic equation.
  • Numerical Methods: Using iterative methods to approximate the solutions.

• Number of Solutions: A cubic equation always has three solutions, but they may be real or complex. It can have:

  • Three distinct real solutions.
  • One real solution and two complex solutions.
  • One real solution with multiplicity three (one unique real solution).

• Unique Solution: A cubic equation has exactly one unique real solution when the other two solutions are complex, or when all three solutions are the same real number. This occurs when the cubic function touches the x-axis at only one point, or when the graph flattens out at the x-axis at a single point.

Exponential Equations

An exponential equation is an equation in which the variable appears in the exponent, such as a^x = b, where a and b are constants.

• Solving Exponential Equations: Exponential equations are typically solved by:

  • Taking Logarithms: Applying logarithms to both sides of the equation to bring the variable down from the exponent.
  • Expressing with the Same Base: Rewriting both sides of the equation with the same base.

• Unique Solution: For an equation of the form a^x = b, where a > 0 and a ≠ 1, there is exactly one solution if b > 0. The solution is x = logₐ(b).

Logarithmic Equations

A logarithmic equation is an equation in which the variable appears within a logarithm, such as logₐ(x) = b, where a and b are constants.

• Solving Logarithmic Equations: Logarithmic equations are typically solved by:

  • Exponentiating: Raising the base of the logarithm to the power of both sides of the equation.
  • Using Logarithmic Properties: Applying properties of logarithms to simplify the equation.

• Unique Solution: For an equation of the form logₐ(x) = b, where a > 0 and a ≠ 1, there is exactly one solution if x > 0. The solution is x = a^b.

Trigonometric Equations

A trigonometric equation is an equation involving trigonometric functions, such as sin(x) = c or cos(x) = c, where c is a constant.

• Solving Trigonometric Equations: Trigonometric equations are solved by:

  • Using Inverse Trigonometric Functions: Applying inverse trigonometric functions to find the principal solutions.
  • Considering Periodicity: Accounting for the periodic nature of trigonometric functions to find all solutions within a given interval.

• Number of Solutions: Trigonometric equations generally have infinitely many solutions due to the periodic nature of trigonometric functions. However, within a specific interval, they may have a finite number of solutions, including exactly one.

  • Example: The equation sin(x) = 1 has exactly one solution x = π/2 in the interval [0, π].

Radical Equations

A radical equation is an equation in which the variable appears under a radical sign, such as √(x) = c, where c is a constant.

• Solving Radical Equations: Radical equations are solved by:

  • Isolating the Radical: Isolating the radical term on one side of the equation.
  • Raising to a Power: Raising both sides of the equation to the power that eliminates the radical.
  • Checking for Extraneous Solutions: Verifying the solutions in the original equation to eliminate extraneous solutions.

• Unique Solution: For an equation of the form √(x) = c, there is exactly one solution if c ≥ 0. The solution is x = c². It's crucial to check this solution in the original equation to ensure it is valid.

Examples of Equations with Exactly One Solution

Let's examine some specific examples to illustrate the concept of equations with exactly one solution.

Example 1: Linear Equation

Consider the equation 5x - 8 = 2.

• Add 8 to both sides: 5x = 10
• Divide both sides by 5: x = 2

This equation has exactly one solution, x = 2.

Example 2: Quadratic Equation

Consider the equation x² - 6x + 9 = 0.

• This can be factored as (x - 3)² = 0
• Therefore, x - 3 = 0
• x = 3

This equation has exactly one solution, x = 3. The discriminant is Δ = (-6)² - 4(1)(9) = 36 - 36 = 0.

Example 3: Exponential Equation

Consider the equation 3^(x+1) = 9.

• Rewrite 9 as 3²: 3^(x+1) = 3²
• Equate the exponents: x + 1 = 2
• x = 1

This equation has exactly one solution, x = 1.

Example 4: Logarithmic Equation

Consider the equation log₂(x - 1) = 3.

• Exponentiate using base 2: 2^(log₂(x - 1)) = 2³
• x - 1 = 8
• x = 9

This equation has exactly one solution, x = 9.

Example 5: Trigonometric Equation

Consider the equation cos(x) = 0 for x ∈ [0, π].

• The cosine function equals 0 at x = π/2 within the given interval.

This equation has exactly one solution in the specified interval, x = π/2.

Example 6: Radical Equation

Consider the equation √(2x + 5) = 3.

• Square both sides: 2x + 5 = 9
• Subtract 5 from both sides: 2x = 4
• Divide by 2: x = 2

Check the solution: √(2(2) + 5) = √(9) = 3. The solution x = 2 is valid.

This equation has exactly one solution, x = 2.

Advanced Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. The solution to a system of equations is the set of values that satisfy all equations simultaneously.

• Linear Systems: A system of linear equations can have:

  • One unique solution (the lines intersect at one point).
  • No solution (the lines are parallel and do not intersect).
  • Infinitely many solutions (the lines are coincident).

• Non-Linear Systems: Non-linear systems can have a variety of solution sets, including one unique solution, multiple solutions, or no solution, depending on the nature of the equations.

Implicit Equations

An implicit equation is an equation in which the dependent variable is not explicitly isolated on one side of the equation, such as x² + y² = 25.

• Solving implicit equations for a specific variable can be complex. In some cases, an implicit equation may define a unique solution for a variable under certain conditions.
  • Example: x² + y² = 25 can be solved for y as y = ±√(25 - x²). For a given x within the domain [-5, 5], there can be two solutions for y, one solution (if x = ±5), or no real solutions (if |x| > 5).

Differential Equations

A differential equation is an equation that relates a function with its derivatives. The solutions to a differential equation are functions rather than numbers.

• Initial Conditions: To find a unique solution to a differential equation, initial conditions or boundary conditions are often required. These conditions specify the value of the function and its derivatives at certain points.
  • Example: The differential equation dy/dx = 2x has the general solution y = x² + C, where C is a constant. If we provide the initial condition y(0) = 1, then 1 = 0² + C, so C = 1, and the unique solution is y = x² + 1.

Practical Applications

Identifying equations with exactly one solution has significant practical applications across various fields:

• Engineering: Solving systems of equations to determine the unique values of parameters in circuit analysis, structural mechanics, and control systems.
• Physics: Finding unique solutions to equations of motion, wave equations, and quantum mechanics.
• Economics: Determining equilibrium points in economic models, where supply equals demand.
• Computer Science: Solving equations in optimization problems, cryptography, and algorithm design.
• Data Science: Determining unique solutions in regression models and machine learning algorithms.

Tips for Identifying Equations with Exactly One Solution

• Understand the Type of Equation: Identify whether the equation is linear, quadratic, exponential, logarithmic, trigonometric, or radical.
• Isolate the Variable: Attempt to isolate the variable on one side of the equation to simplify the solution process.
• Use Appropriate Methods: Apply appropriate methods for solving each type of equation, such as factoring, completing the square, quadratic formula, logarithms, exponentiation, or trigonometric identities.
• Check the Discriminant: For quadratic equations, use the discriminant to determine the number of real solutions.
• Consider the Domain: Account for the domain of the variable, especially for logarithmic, radical, and trigonometric equations.
• Check for Extraneous Solutions: Verify the solutions in the original equation, particularly for radical and logarithmic equations.
• Use Graphical Methods: Graph the equation to visually determine the number of solutions and identify the unique solution.
• Apply Numerical Methods: Use numerical methods to approximate the solutions when analytical methods are difficult.

Conclusion

Identifying equations with exactly one solution is a fundamental skill in mathematics, with broad applications across various disciplines. By understanding the nature of different types of equations and applying appropriate solution techniques, you can effectively determine the unique values that satisfy these equations. Whether you're solving linear, quadratic, exponential, logarithmic, trigonometric, or radical equations, a systematic approach and careful consideration of the domain and potential extraneous solutions will lead to accurate results. This skill is not only valuable in academic settings but also essential for practical problem-solving in engineering, physics, economics, computer science, and beyond. Embrace the power of equations and unlock the secrets hidden within their solutions.