Solving a quadratic equation is only half the job. The other half is making sure your answers are correct. Many students skip this step, but checking your work is important. It helps you catch small errors before they turn into big ones.
In this guide, you’ll learn how to check quadratic solutions using simple and reliable methods.
Use Substitution
The most direct way to check your solution is by plugging the values back into the original equation.
Step-by-step
-
Solve the equation
-
Take each solution
-
Substitute it into the original equation
-
See if the left-hand side equals the right-hand side
Example
Solve: x² – 5x + 6 = 0
Factored form: (x – 2)(x – 3) = 0
Solutions: x = 2 and x = 3
Check x = 2
Left side: (2)² – 5(2) + 6 = 4 – 10 + 6 = 0
Right side: 0
It checks out
Check x = 3
Left side: (3)² – 5(3) + 6 = 9 – 15 + 6 = 0
This is correct too
If both values make the equation true, your answers are correct.
Use Graphing
Graphing is another way to confirm your solutions. A quadratic equation creates a U-shaped curve called a parabola. The points where the curve touches the x-axis are the solutions or roots.
You can graph the equation manually, with a calculator, or using an online graphing tool like Desmos.
Steps
-
Rewrite the equation as y = ax² + bx + c
-
Graph it
-
Find the x-values where y = 0
-
Compare those x-values with your solutions
Example
Equation: x² – 4x – 5 = 0
Solutions: x = 5 and x = -1
Graph y = x² – 4x – 5
The parabola crosses the x-axis at x = 5 and x = -1
So the solutions are correct
Graphing is especially helpful if your answers are decimals or irrational numbers. You can estimate where the curve crosses the x-axis to check your work.
Use the Discriminant
The discriminant tells you what kind of solutions to expect. It doesn’t confirm the exact values, but it helps you know if the type of solution you found makes sense.
The discriminant is the part under the square root in the quadratic formula:
b² – 4ac
What it tells you
-
If it’s positive, expect two real solutions
-
If it’s zero, expect one real solution
-
If it’s negative, expect two complex solutions
Example
Equation: x² + 2x + 5 = 0
Discriminant: 2² – 4(1)(5) = 4 – 20 = -16
You should get complex solutions
If you find real roots, something is wrong
While it’s not a full check, it confirms whether your solution type matches the math.
Estimate to Catch Mistakes
Sometimes, rough estimation can help you see if a solution is reasonable.
For example, if your answer is x = 50 but the equation is x² – 2x + 1 = 0, that doesn’t make sense. A large number like 50 would result in a much larger output. So x = 50 is probably wrong.
If you’re solving x² – 4 = 0 and your solution is x = 10, try substituting it quickly:
10² – 4 = 100 – 4 = 96, not 0
This tells you the answer is incorrect.
Estimation is fast and can point out large mistakes even before checking step by step.
Final Thoughts
Knowing how to check quadratic solutions helps you avoid simple errors. Use substitution for exact verification. Use graphing to confirm visually. Use the discriminant to double-check the type of solution. And use estimation when you’re short on time.
These tools work together to make sure your answers are not only complete, but correct. Always check your work—it’s a habit that builds accuracy and confidence in math.
