Completing the Square Calculator
Complete the square for any quadratic ax²+bx+c. Get vertex form, vertex coordinates, axis of symmetry, discriminant, roots, and step-by-step solution.
Completing the Square Practice
Practice completing the square with generated problems. Enter your answer for h and k, check if correct, and get step-by-step solutions with multiple difficulty levels.
Problem: x² + 6x + 5
Complete the square to find h and k in a(x − h)² + k
Your Answer:
Score⧉
0 / 0 correct (0%)
Your running accuracy across problems
Planning notes, formulas, and examples
About the Completing the Square Practice
Mastering completing the square requires practice — lots of it. This interactive page generates quadratic expressions and challenges you to find the vertex coordinates h and k. You enter your answers, and the calculator checks them with a step-by-step solution so you can see exactly where you went right or wrong.
Three difficulty levels let you progress at your own pace. Easy mode keeps the leading coefficient at 1 with small values of b and c — perfect for beginners. Medium mode introduces leading coefficients up to 3 and larger values. Hard mode throws in negative leading coefficients and a wide range of values that mirror what you might see on an algebra exam.
Each problem comes with a generate button that creates random coefficients matching your chosen difficulty. A running score tracks your accuracy across multiple problems, so you can set goals like "10 in a row" before moving up a level. After submitting, you see the full step-by-step solution and accuracy bars showing how close your answers were. A reference table of worked examples provides additional study material. This is the ideal companion page for students preparing for algebra tests, SAT math sections, or anyone who wants to build algebraic fluency through deliberate practice.
When This Page Helps
Use this when you want repeated completing-the-square problems with immediate worked feedback instead of solving isolated examples by hand. It is useful for exam prep and tutoring because the coefficients, completed form, and checking steps stay tied to the same practice question.
How to Use the Inputs
- Choose a difficulty level: Easy, Medium, or Hard.
- Click "Generate New Problem" or use a preset to load a quadratic.
- Work out h and k on paper or in your head.
- Enter your h and k values in the answer section.
- Click "Check Answer" to see if you are correct.
- Review the step-by-step solution and try another problem.
Formula used
Given ax² + bx + c:
h = −b / (2a)
k = c − b² / (4a)
Vertex form: a(x − h)² + kExample Calculation
Result: For difficulty=5, coefficienta=10, coefficientb=15, the tool returns the solved completing the square practice outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in completing the square practice formulas and reports derived values, checks, and classifications automatically.
Tips & Best Practices
- Always start by identifying a, b, and c from the standard form.
- For easy mode: h = −b/2 and k = c − b²/4 since a = 1.
- Double-check your sign: h = −b/(2a), so if b is positive, h is negative.
- Practice with a = 1 first until you can do it quickly, then move to a ≠ 1.
- The worked examples table is an excellent reference — study the patterns.
When To Use This Calculator
Practice completing the square with generated problems. Enter your answer for h and k, check if correct, and get step-by-step solutions with multiple difficulty levels. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
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Frequently Asked Questions
The coefficients a, b, and c are randomly generated within the range defined by your chosen difficulty level. Easy: a = 1; Medium: a up to 3; Hard: a can be negative, wider ranges.
Your answer must be within 0.01 of the correct value. For exact integer answers, type the integer. For fractions, use the decimal form.
Enter the decimal equivalent. For example, for h = −3/2, enter −1.5.
The score resets when you reload the page. It tracks your accuracy within the current practice session.
The generator ensures a is never 0, since that would not be a quadratic expression.
Absolutely. The SAT frequently tests completing the square. Start on Easy, work up to Hard, and aim for consistent accuracy before moving on.
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