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Completing the Square Calculator: Step-by-Step Solutions

Completing the Square Calculator: Step-by-Step Solutions

How to Complete the Square for Any Quadratic Equation

Completing the Square Calculator – Free Online Tool with Formula, Stepwise Method & Examples

Completing the Square Calculator

Enter coefficient a (x²): Enter coefficient b (x): Enter constant c:

What is Completing the Square Calculator?

The Completing the Square Calculator is an easy online tool for solving quadratic equations of the form ax² + bx + c = 0 using the completing the square method. This technique rewrites the quadratic equation in a form that makes it simple to find the roots (solutions) and to convert the equation to its vertex form. It provides step-by-step transformations, making abstract algebra easier to visualize and understand.

Formula or Logic Behind Completing the Square Calculator

Completing the square involves a mathematical process that transforms a quadratic equation into a perfect square trinomial, allowing you to solve for x. The standard formula is:

ax² + bx + c = 0
⇒ x² + (b/a)x = -c/a
Add (b/2a)² to both sides to complete the square:
x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²

Which simplifies into the square form:
(x + b/2a)² = (b² - 4ac)/4a²

Solving for x gives the roots using the steps shown by the calculator above.

Completed Square Form & Solved Examples

a	b	c	Completed Square Form	Roots
1	6	5	(x + 3)² - 4 = 0	x = -5, -1
2	-8	6	2(x - 2)² - 2 = 0	x = 1, 3
1	4	1	(x + 2)² - 3 = 0	x = -2 ± √3
1	2	-8	(x + 1)² = 9	x = 2, -4

Steps to Use the Completing the Square Calculator

Enter the required coefficients a, b, and c for your quadratic equation.
Click on the 'Calculate' button.
Get instant results, including the completed square form, detailed solution steps, and the final roots.

Why Use Vedantu’s Completing the Square Calculator?

Vedantu's Completing the Square Calculator is designed to be user-friendly, mobile-compatible, and perfect for quick algebra help. It details every calculation step, helping students, teachers, and self-learners easily master this quadratic solving method. Trusted by lakhs of students, it follows curriculum guidelines and is ideal for homework, board exams, JEE/NEET, and beyond.

Real-life Applications of Completing the Square

The completing the square process is used in many real-world scenarios, such as: finding the vertex of a parabola for graphing, solving physics equations about projectile motion and trajectories, optimizing areas and profits in business problems, and in calculus for integrating quadratic expressions. In academics, it helps transition to higher-level algebra and calculus quickly.

You can use this calculator when studying quadratic equations, checking homework, or interpreting parabolic graphs in sciences.

For related maths calculators and algebra topics, explore: Prime Numbers, Quadratics, Factors, HCF Calculator, and Algebra Topics.

FAQs on Completing the Square Calculator: Step-by-Step Solutions

1. What is completing the square?

Completing the square is an algebraic method used to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, making it easier to find the roots (solutions) of the quadratic equation. This method is particularly useful when factoring is difficult or impossible.

2. What is the formula for completing the square?

The general formula is based on the perfect square trinomial pattern (a + b)² = a² + 2ab + b². For a quadratic equation in the form ax² + bx + c = 0, completing the square involves manipulating the equation to resemble this pattern. The specific steps depend on the equation’s coefficients, but the core idea is to make a perfect square trinomial.

3. How do you solve quadratic equations by completing the square?

Solving quadratic equations by completing the square involves these steps: 1. Make sure the coefficient of x² is 1. 2. Move the constant term to the right side of the equation. 3. Take half of the coefficient of x, square it, and add it to both sides. 4. Factor the left side as a perfect square. 5. Take the square root of both sides, and solve for x.

4. What are the steps to completing the square, step-by-step?

The steps are: 1. Ensure the coefficient of x² is 1. 2. Move the constant term to the right side. 3. Find half of the coefficient of x, square it, and add to both sides. 4. Factor the left side into a perfect square. 5. Take the square root of both sides and solve for x. This method converts the quadratic into vertex form, which reveals the vertex of the parabola.

5. What is the completing the square method used for?

The completing the square method is primarily used to solve quadratic equations, especially when factoring is not straightforward. It's also crucial for converting a quadratic equation into vertex form, which readily provides the coordinates of the parabola's vertex.

6. How is completing the square related to the quadratic formula?

The quadratic formula is actually derived from the process of completing the square on the general quadratic equation ax² + bx + c = 0. Completing the square provides a way to solve quadratics without memorizing the quadratic formula, although the formula itself provides a quicker shortcut for many.

7. When should I use completing the square versus the quadratic formula?

Use completing the square when you need to understand the process of transforming the equation to vertex form. This is helpful in revealing the vertex of the parabola. The quadratic formula is generally faster for direct calculation of roots. Both methods are equally valid for solving for x.

8. What are some examples of completing the square?

Let's solve x² + 6x + 5 = 0 using completing the square: 1. Move the constant: x² + 6x = -5. 2. Half of 6 is 3; 3² = 9; add 9 to both sides: x² + 6x + 9 = 4. 3. Factor the left side: (x + 3)² = 4. 4. Take the square root: x + 3 = ±2. 5. Solve for x: x = -1 or x = -5. Other examples involve equations with different 'a', 'b', and 'c' values, always following the same fundamental process.

9. Is completing the square always the best method to solve quadratics?

No, completing the square is not always the best method. For simple quadratics, factoring might be faster and easier. The quadratic formula is generally quicker for complex equations, especially when the solutions are irrational numbers. The best method depends on the specific equation and your familiarity with different techniques.

10. Why is completing the square important in mathematics?

Completing the square is a fundamental algebraic technique with wide applications. Besides solving quadratic equations, it's crucial in transforming quadratics into vertex form, which allows for easy identification of the parabola’s vertex and axis of symmetry. This transformation is essential in various areas such as conic sections, calculus, and other advanced mathematical concepts.

11. What does b/2a represent in completing the square?

In the process of completing the square for a quadratic equation of the form ax² + bx + c = 0, b/2a represents half of the coefficient of the x term. Squaring this value ( (b/2a)² ) and adding it to both sides of the equation creates the perfect square trinomial. This value is essential in forming the completed square and subsequently finding the vertex of the parabola represented by the quadratic equation.

12. Can completing the square be used to solve any quadratic equation?

Yes, completing the square can be used to solve any quadratic equation, even those that are difficult or impossible to factor. However, for equations with complex coefficients or where a quick solution is desired, the quadratic formula might be a more efficient approach. But completing the square always works as a method to find the roots.