Solution Concept Formula

The Formula

If f(a) = 0, then x = a is a solution of f(x) = 0

When to use: The answer to 'what value of x makes this equation true?' — found by solving, confirmed by checking.

Quick Example

x = 5 is a solution to 2x + 3 = 13 because 2(5) + 3 = 13

Notation

A solution is written x = a. Verification uses substitution: replace x with a and check both sides are equal.

What This Formula Means

A specific value (or set of values) that makes an equation or inequality true when substituted in for the variable.

The answer to 'what value of x makes this equation true?' — found by solving, confirmed by checking.

Formal View

A value a \in D is a solution of the equation f(x) = g(x) iff f(a) = g(a), i.e., a \in \{x \in D \mid f(x) = g(x)\}.

Worked Examples

Example 1

easy
Is x = 3 a solution of x^2 - 9 = 0?

Solution

  1. 1
    Substitute x = 3: (3)^2 - 9 = 9 - 9 = 0.
  2. 2
    Since the result is 0, x = 3 satisfies the equation.
  3. 3
    Therefore x = 3 is a solution.

Answer

Yes, x = 3 is a solution.
A solution is a value that, when substituted into the equation, makes the equation true. The verification process is called checking the solution.

Example 2

medium
Find all solutions of |x| = 5.

Common Mistakes

  • Declaring a value is a solution without substituting it back into the original equation to verify
  • Stopping after finding one solution when the equation has multiple solutions (e.g., x^2 = 4 has two)
  • Confusing an intermediate step value with the final solution

Why This Formula Matters

Understanding solutions precisely defines what 'solving' means — finding every value that makes the equation true.

Frequently Asked Questions

What is the Solution Concept formula?

A specific value (or set of values) that makes an equation or inequality true when substituted in for the variable.

How do you use the Solution Concept formula?

The answer to 'what value of x makes this equation true?' — found by solving, confirmed by checking.

What do the symbols mean in the Solution Concept formula?

A solution is written x = a. Verification uses substitution: replace x with a and check both sides are equal.

Why is the Solution Concept formula important in Math?

Understanding solutions precisely defines what 'solving' means — finding every value that makes the equation true.

What do students get wrong about Solution Concept?

Always verify: substitute your answer back into the original equation to confirm both sides are equal.

What should I learn before the Solution Concept formula?

Before studying the Solution Concept formula, you should understand: equations.

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