Will a one-variable equation always have only one solution?

Steps to Solve

1. Understand the types of one-variable equations

One-variable equations can be linear equations (e.g., $ax + b = 0$) or quadratic equations (e.g., $ax^2 + bx + c = 0$). The nature of these equations can determine the number of solutions.

2. Examine linear equations

A linear equation typically has the form $ax + b = 0$. When you solve for $x$, you will generally find one solution unless $a = 0$, which would lead to no solutions or infinitely many (if also $b = 0$).

3. Examine quadratic equations

Quadratic equations take the form $ax^2 + bx + c = 0$. The number of solutions depends on the discriminant ($D = b^2 - 4ac$):

• If $D > 0$: two distinct solutions.
• If $D = 0$: one solution (a repeated root).
• If $D < 0$: no real solutions.

4. Illustrate with examples

Consider the linear equation $2x - 4 = 0$. Solving gives: $$ 2x = 4 \implies x = 2 $$ That's one solution.

For the quadratic equation $x^2 - 4x + 4 = 0$, the discriminant is $b^2 - 4ac = 0$. Thus there's one repeated solution: $$ x = \frac{-b}{2a} = \frac{4}{2} = 2 $$.

5. Conclusion on one-variable equations

Thus, a one-variable equation does not always have one solution. It can have zero, one, or multiple solutions depending on its type and specific coefficients.