Which Of The Following Equations Have Exactly One Solution

Which of the following equations have exactly one solution?
Finding the exact number of solutions for an equation is a fundamental skill in algebra and calculus, and it helps students understand the behavior of functions and the nature of mathematical relationships. In this article, we will explore the conditions that guarantee a single solution, examine several common types of equations, and walk through illustrative examples that show how to determine whether an equation has one, none, or infinitely many solutions.

Introduction

When we ask whether an equation has exactly one solution, we are looking for a unique value that satisfies the equation’s conditions. This concept appears in many contexts—from linear equations that intersect at a single point, to nonlinear equations that curve and cross the x‑axis only once. Understanding the underlying principles allows you to solve equations efficiently and predict the behavior of more complex systems.

1. Types of Equations and Their Solution Sets

Equation Type	Typical Form	Typical Solution Set
Linear	(ax + b = 0)	One solution if (a \neq 0); none if (a = 0) and (b \neq 0); infinitely many if (a = 0) and (b = 0).
Quadratic	(ax^2 + bx + c = 0)	0, 1, or 2 real solutions depending on the discriminant (D = b^2 - 4ac).
Cubic	(ax^3 + bx^2 + cx + d = 0)	1 or 3 real solutions; multiplicities matter.
Rational	(\frac{P(x)}{Q(x)} = 0)	Solutions where (P(x)=0) and (Q(x)\neq 0).
Transcendental	(f(x) = g(x)) where (f, g) involve exponentials, logs, trig	Typically solved numerically; uniqueness depends on monotonicity.
Systems of Equations	Multiple equations in multiple variables	Unique solution if the system is consistent and independent.

The key to determining uniqueness lies in the properties of the functions involved: linearity, monotonicity, continuity, and the presence of constraints (like domain restrictions).

2. Criteria for a Single Solution

Below are the main mathematical criteria that guarantee exactly one solution.

2.1 Linear Equations

A linear equation in one variable, (ax + b = 0), has a unique solution when (a \neq 0). The solution is simply (-b/a).

2.2 Quadratic Equations

For (ax^2 + bx + c = 0) with (a \neq 0), the discriminant (D = b^2 - 4ac) determines the number of real solutions:

(D > 0): Two distinct real solutions.
(D = 0): Exactly one real solution (a double root).
(D < 0): No real solutions (complex solutions only).

Thus, a quadratic has exactly one real solution when (D = 0).

2.3 Cubic and Higher‑Degree Polynomials

A cubic equation has a unique real solution when the discriminant is negative, indicating one real root and two complex conjugates. For higher‑degree polynomials, the Fundamental Theorem of Algebra guarantees at least one real root, but uniqueness depends on the function’s shape and derivative behavior. A useful rule of thumb: if the polynomial’s derivative has no real roots (i.e., the function is strictly monotonic), the polynomial is one‑to‑one and thus has exactly one real root.

2.4 Rational Equations

A rational equation (\frac{P(x)}{Q(x)} = 0) reduces to solving (P(x) = 0) while ensuring (Q(x) \neq 0). If (P(x)) is linear and (Q(x)) has no real zeros that coincide with the zero of (P(x)), the solution is unique.

2.5 Transcendental Equations

For equations like (e^x = 2x) or (\sin x = x/2), uniqueness often hinges on monotonicity. If the function (f(x) = e^x - 2x) is strictly increasing (its derivative (f'(x) = e^x - 2 > 0) for all real (x)), it crosses the x‑axis only once, guaranteeing a single solution.

If you found this helpful, you might also enjoy wrapping a foot with plantar fasciitis or words that start with s and end with d.

2.6 Systems of Equations

A system of (n) equations in (n) variables has a unique solution if the coefficient matrix is invertible (determinant non‑zero). In geometric terms, the corresponding planes or lines intersect at exactly one point.

3. Step‑by‑Step Procedure to Test for a Unique Solution

Identify the Equation Type – Linear, quadratic, polynomial, rational, transcendental, or a system.
Simplify – Reduce to standard form, factor if possible.
Check Coefficients – For linear or rational equations, ensure the leading coefficient is non‑zero.
Compute the Discriminant (if applicable) – For quadratics, calculate (D = b^2 - 4ac).
Analyze Monotonicity – Determine if the function is strictly increasing or decreasing by examining the derivative.
Check Domain Restrictions – Exclude values that make denominators zero or arguments of logs negative.
Apply the Criteria – Use the rules from Section 2 to decide if exactly one solution exists.
Verify – Substitute the solution back into the original equation to confirm it satisfies all constraints.

4. Illustrative Examples

Example 1: Linear Equation

[ 5x - 10 = 0 ]

Leading coefficient (a = 5 \neq 0).
Unique solution: (x = 2).
Verification: (5(2) - 10 = 0).

Example 2: Quadratic Equation

[ x^2 - 4x + 4 = 0 ]

Discriminant (D = (-4)^2 - 4(1)(4) = 16 - 16 = 0).
Exactly one real solution: (x = 2) (double root).
Verification: ((2)^2 - 4(2) + 4 = 0).

Example 3: Cubic Equation

[ x^3 - 3x + 2 = 0 ]

Factor: ((x+1)(x^2 - x - 2) = (x+1)(x+1)(x-2)).
Roots: (x = -1) (double root), (x = 2).
Two distinct real solutions: not unique.

Example 4: Rational Equation

[ \frac{x - 3}{x + 1} = 0 ]

Numerator zero: (x = 3).
Denominator non‑zero at (x = 3).
Unique solution: (x = 3).