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120 LearnersLast updated on September 2, 2025
Completing The Square Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like trigonometry. Whether you're cooking, tracking BMI, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about completing the square calculators.
What is a Completing The Square Calculator?
A completing the square calculator is a tool to solve quadratic equations by converting them into a perfect square trinomial. This method can simplify solving equations and finding the roots. The calculator makes this process much easier and faster, saving time and effort.
How to Use the Completing The Square Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the quadratic equation: Input the equation in the given field.
Step 2: Click on solve: Click on the solve button to complete the square and get the result.
Step 3: View the result: The calculator will display the completed square form and roots instantly.
How to Complete the Square?
To complete the square for a quadratic equation in the form ax2 + bx + c, you can follow these steps:
1. Divide all terms by 'a' if 'a' is not 1.
2. Rearrange the equation to isolate x2 + (b/a)x.
3. Add and subtract (b/2a)2 to complete the square.
4. Factor the perfect square trinomial. The formula used is: (x + b/2a)2 - (b/2a)2 + c/a = 0.
Tips and Tricks for Using the Completing The Square Calculator
When using a completing the square calculator, there are a few tips and tricks to make it easier and avoid mistakes:
Understand the purpose of completing the square, which is to simplify solving the equation.
Keep track of all transformations applied to the equation.
Use the calculator's output to verify your manual calculations.
Common Mistakes and How to Avoid Them When Using the Completing The Square Calculator
We may think that when using a calculator, mistakes will not happen. But it is possible for children to make mistakes when using a calculator.
Mistake 1
Failing to Divide by 'a' When Necessary
Ensure that the coefficient of x2 is 1 before completing the square. If a ≠ 1, divide through by 'a' first to avoid errors.
Mistake 2
Incorrectly Calculating (b/2a)2
Make sure to calculate (b/2a)2 accurately, as this value is critical in completing the square.
Mistake 3
Forgetting to Add and Subtract the Same Value
Remember to add and subtract the same value (b/2a)2 to maintain equation balance. Omitting this step leads to incorrect results.
Mistake 4
Misinterpreting the Completed Square Form
The completed square form should be interpreted correctly to find the roots. Use the form (x + p)2 = q to solve for x.
Mistake 5
Over-reliance on Calculator Output
Check calculator results against manual calculations for accuracy, especially in exams or critical applications.
Completing The Square Calculator Examples
Problem 1
Solve the quadratic equation x^2 + 6x + 5 using completing the square.
Step 1: Rearrange the equation: x2 + 6x = -5
Step 2: Add and subtract (6/2)2: x2 + 6x + 9 = 4
Step 3: Factor: (x + 3)2 = 4
Step 4: Solve: x + 3 = ±2 x = -1 or x = -5
Explanation
By completing the square, we factor the equation into (x + 3)2 = 4, allowing us to solve for x easily.
Problem 2
Use completing the square to solve 2x^2 + 8x - 10 = 0.
Step 1: Divide by 2: x2 + 4x = 5
Step 2: Add and subtract (4/2)2: x2 + 4x + 4 = 9
Step 3: Factor: (x + 2)2 = 9
Step 4: Solve: x + 2 = ±3 x = 1 or x = -5
Explanation
Dividing through by 2 simplifies the equation, and completing the square gives (x + 2)2 = 9.
Problem 3
Find the roots of 3x^2 - 12x + 9 using completing the square.
Step 1: Divide by 3: x2 - 4x = -3
Step 2: Add and subtract (4/2)2: x2 - 4x + 4 = 1
Step 3: Factor: (x - 2)2 = 1
Step 4: Solve: x - 2 = ±1 x = 3 or x = 1
Explanation
By completing the square, we transform the equation to (x - 2)2 = 1, which is straightforward to solve.
Problem 4
Solve x^2 - 10x + 16 = 0 by completing the square.
Step 1: Rearrange: x2 - 10x = -16
Step 2: Add and subtract (10/2)^2: x2 - 10x + 25 = 9
Step 3: Factor: (x - 5)2 = 9
Step 4: Solve: x - 5 = ±3 x = 8 or x = 2
Explanation
Completing the square results in (x - 5)2 = 9, making it easy to find the roots.
Problem 5
Use completing the square for 4x^2 + 12x + 9 = 0.
Step 1: Divide by 4: x2 + 3x = -2.25
Step 2: Add and subtract (3/2)2 = 2.25: x2 + 3x + 2.25 = 0
Step 3: Factor: (x + 1.5)2 = 0
Step 4: Solve: x + 1.5 = 0 x = -1.5
Explanation
After dividing and completing the square, we find (x + 1.5)2 = 0, leading to a single root.
FAQs on Using the Completing The Square Calculator
1.How do you complete the square for a quadratic equation?
2.When is completing the square useful?
3.Can completing the square be used for all quadratics?
4.Is completing the square the same as using the quadratic formula?
5.What is the benefit of using a completing the square calculator?
Glossary of Terms for the Completing The Square Calculator
- Quadratic Equation: An equation of the form ax2 + bx + c = 0.
- Perfect Square Trinomial: A trinomial that can be factored into a binomial squared, e.g., (x + p)2.
- Vertex Form: A way of expressing a quadratic equation as a(x - h)2 + k.
- Roots: The solutions to a quadratic equation where it equals zero.
- Factor: To express an equation as a product of its factors, such as a binomial.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
