Verify that x = 3 is a solution of
the equation x2 – 5x +
6 = 0.
To do this, substitute x = 3 into the equation.
32 – 5·3 +
6 = 0? 9 – 15 + 6 = 0?
0
= 0 confirmed
Key Formula
Substitute the candidate value into the original equation and check: f(x0)=?g(x0)
Where:
x0 = The proposed solution you want to verify
f(x0) = The left-hand side of the equation evaluated at x₀
g(x0) = The right-hand side of the equation evaluated at x₀
Worked Example
Problem: Verify that x = 5 is a solution of the equation 2x + 3 = 13.
Step 1: Write down the original equation.
2x+3=13
Step 2: Substitute x = 5 into the left-hand side.
2(5)+3=10+3=13
Step 3: Compare the result with the right-hand side.
13=13✓
Step 4: Since both sides are equal, the candidate value satisfies the equation.
Answer: x = 5 is verified as a solution of 2x + 3 = 13.
Another Example
This example differs from the first because it verifies a solution to an inequality rather than an equation. You check that the inequality symbol holds, not just that both sides are equal.
Problem: Verify whether x = −2 is a solution of the inequality 3x − 1 < 4.
Step 1: Write down the original inequality.
3x−1<4
Step 2: Substitute x = −2 into the left-hand side.
3(−2)−1=−6−1=−7
Step 3: Check whether the inequality holds.
−7<4✓
Step 4: The left-hand side is indeed less than 4, so the inequality is satisfied.
Answer: x = −2 is verified as a solution of the inequality 3x − 1 < 4.
Frequently Asked Questions
What is the difference between solving and verifying a solution?
Solving means finding the value(s) of the variable that make the equation or inequality true. Verifying means taking a value you already have and substituting it back into the original statement to confirm it works. Solving is the discovery step; verifying is the confirmation step.
Why is it important to verify a solution?
Algebra can introduce errors at any step—sign mistakes, arithmetic slips, or even extraneous solutions that arise from squaring both sides or clearing denominators. Verifying catches these errors by going back to the original problem. It is especially critical in radical equations and rational equations where extraneous solutions commonly appear.
How do you verify a solution to a system of equations?
Substitute the proposed values into every equation in the system, not just one. Each equation must be satisfied. If even one equation fails, the candidate is not a solution to the system.
Verify a Solution vs. Solve an Equation
Verify a Solution
Solve an Equation
Goal
Confirm that a given value is correct
Find the unknown value(s)
Starting point
You already have a candidate answer
You start with only the equation
Method
Substitute and check both sides
Use algebraic operations to isolate the variable
Result
"Confirmed" or "Not a solution"
A value or set of values (e.g., x = 3)
When to use
After solving, or when given a candidate to test
When the problem asks you to find the answer
Why It Matters
Verifying solutions is a standard step in algebra, geometry proofs, and standardized tests. Many teachers require it as the final step of any problem, and showing verification can earn you partial or full credit even if your solving process had a minor error. In topics like radical equations and logarithmic equations, verification is essential because algebraic manipulation can produce extraneous (false) solutions that must be identified and rejected.
Common Mistakes
Mistake: Substituting into a simplified or rearranged form of the equation instead of the original.
Correction: Always substitute back into the original equation or inequality as it was first given. A rearranged equation may have introduced errors that your verification would miss if you use it as the check.
Mistake: Checking only one equation when verifying a solution to a system of equations.
Correction: A solution to a system must satisfy all equations simultaneously. Substitute into every equation in the system and confirm each one holds.
