Question
Give me a few equations that have no solution
Answers
GPT-4o mini
Certainly! Here are a few equations that have no solutions:
1. **Linear Equation with Contradiction**:
\[
2x + 3 = 2x + 5
\]
Subtracting \(2x\) from both sides gives \(3 = 5\), which is a contradiction.
2. **Quadratic Equation with No Real Roots**:
\[
x^2 + 4x + 5 = 0
\]
The discriminant (\(b^2 - 4ac\)) is \(4 - 20 = -16\), which is negative, indicating no real solutions.
3. **Rational Equation**:
\[
\frac{1}{x} = \frac{1}{(x-1)} \quad \text{for } x \neq 0, 1
\]
Cross-multiplying gives \(1(x-1) = 1x\) or \(x - 1 = x\), leading to \( -1 = 0\), which is a contradiction.
4. **Absolute Value Equation**:
\[
|x - 2| = -3
\]
Since the absolute value cannot be negative, this equation has no solution.
5. **Trigonometric Equation**:
\[
\sin(x) = 2
\]
The sine function only outputs values in the range \([-1, 1]\), so there are no solutions for this equation.
These examples illustrate different types of equations with no solutions due to contradictions or inherent mathematical properties.
1. **Linear Equation with Contradiction**:
\[
2x + 3 = 2x + 5
\]
Subtracting \(2x\) from both sides gives \(3 = 5\), which is a contradiction.
2. **Quadratic Equation with No Real Roots**:
\[
x^2 + 4x + 5 = 0
\]
The discriminant (\(b^2 - 4ac\)) is \(4 - 20 = -16\), which is negative, indicating no real solutions.
3. **Rational Equation**:
\[
\frac{1}{x} = \frac{1}{(x-1)} \quad \text{for } x \neq 0, 1
\]
Cross-multiplying gives \(1(x-1) = 1x\) or \(x - 1 = x\), leading to \( -1 = 0\), which is a contradiction.
4. **Absolute Value Equation**:
\[
|x - 2| = -3
\]
Since the absolute value cannot be negative, this equation has no solution.
5. **Trigonometric Equation**:
\[
\sin(x) = 2
\]
The sine function only outputs values in the range \([-1, 1]\), so there are no solutions for this equation.
These examples illustrate different types of equations with no solutions due to contradictions or inherent mathematical properties.