Completing the Square

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Completing the square is a method used to determine roots of a given quadratic equation. Any polynomial equation with a degree that is equal to 2 is known as quadratic equations. Even though ‘quad’ means four, but ‘quadratic’ represents ‘to make square’. The standard form of representing a quadratic equation is, ay² + by + c = 0, where a, b and c are real numbers, where a is not equal to 0 and y is a variable.


Moreover, since this degree of this above-mentioned equation is 2, then it will contain two roots or solutions. Roots of polynomials represent different values of x that ultimately satisfy this equation. Apart from that, there are various methods to determine the root of a quadratic equation. One of the prominent ones here is completing the square method. Students need to learn this fundamental to understand advanced concepts related to this section of Mathematics.


Solving Quadratic Equations by Completing the Square

Before starting this process, one needs to identify a suitable equation for it, here is one -ax2+bx+c= 0. Now to solve this equation via this process, here are the essential to completing the square steps –

  1. At first, transform this equation in a way so that this constant term, i.e. ‘c’ remains on the right side of an equation.

  2. Now, if ‘a’ the leading coefficient (coefficient of x2 term) is not equal to 1, then divide both sides via a.

  3. After that, add the square of half of the coefficient of ‘x’ (b/2a)2 to both sides of an equation.

  4. Following that, consider the left side of an equation as the square of a binomial.

  5. Then, take the square root of each side.

  6. Find the solution for x.

One can also solve a quadratic equation by completing the square method using geometry. To do that, a perfect way would be to represent the terms of expression in the L.H.S of an equation. Students can use geometric figures like squares, rectangles, etc. Now if one takes a square with sides equal to x units, then it will have an area of x2 units. This will represent the first term of expression. Similarly, a rectangle with sides a and b will have an area of ‘ab’ square units. This method is known as completing the square.


Square Root Property

When there are no linear terms in an equation, another way of solving a quadratic equation is using the square root property. Here students will isolate the x2 term and take its square root value on the other side of an equal sign. However, one must remember that at times one needs to manipulate this equation to perform this isolation of x2 to use this method.


Property of Square Root

With the isolation of x2, the property of this method suggests that, 

if x2​= k, in that case, x=±√​k​​​.


Examples to Solve By Completing the Square

Solving x2 – 6x – 3= 0 by using completing square method formula –

  • x2 – 6x – 3 = 0

  • x2 – 6x = 3

  • x2−6x+(−3)2 = 3+9

  • (x−3)2 = 12

  • x−3 = ± √12 = ±2 √3

  • x = 3±2√3

Completing the square allows students a way to solve any quadratic equation without many difficulties. Moreover, in case of any larger equations, this method proves fruitful. Therefore, it is vital for students to learn and comprehend this concept and its associated ideas to solve equations without any difficulties.


Furthermore, with online learning platforms like Vedantu, it is easy to comprehend such complicated concepts. Study materials with easy explanations, lucid language and various real-life examples help students to improve their preparations. Moreover, the online live classes and doubt clearing session helps further in this process.

FAQ (Frequently Asked Questions)

1. Who is the Father of Completing the Square Method?

Ans. Muhammed ibn Musa al-Khwarizmi or Al-Khwarizmi is regarded as the father of completing the square equation. Owing to the command of his Caliph, he arranged various materials on algebra and prepared his first text. After the initial introduction and first-degree equations and simple quadratics, which require square root as its solution, he shifted to quadratic equations. He never uses the term ‘quadratic equation’ but focuses on the dimension of a square and solves it via the method used today.

2. Who Coined the Term Quadratic Formula?

Ans. The history of the quadratic formula can be traced back to ancient Egypt. However, in recent time, Persian mathematician, Al-Khwarizmi first solved this equation algebraically. Moreover, in 1594, Simon Stevin first obtained a quadratic equation covering all cases, and in 1637 René Descartes published his works in ‘La Géométrie’. It contained the special cases of a quadratic equation as popularly known today.

3. What is the Usage of a Quadratic Equation?

Ans. A quadratic equation helps to find the curve of a Cartesian grid. It helps to determine the curve that an object takes while flying through the air. For instance, while designing a football, basketball, cricket ball, etc., this pointer plays a part. Moreover, it also helps to determine the shape of any object that requires a specified curved shape. For instance, while producing missiles, scientists use this formula to determine where they want it to hit.

4. What is the Connection Between Newton’s Law of Motion and the Quadratic Equation?  

Ans. Newton constructed his law of motion on quadratic equations. He used this formula to define the acceleration of objects and forces. His law focuses on the movement and fall of objects, considering the rotation of the earth. However, Newton was not aware of the forces that work within the solar system owing to the rotation of the Milky Way Galaxy.