Completing the square method is one of the methods to find the roots of the given quadratic equation. A polynomial equation with degree equal to two is known as a quadratic equation. ‘Quad’ means four but ‘Quadratic’ means ‘to make square’. A quadratic equation in its standard form is represented as:
ax2 + bx + c = 0, where a,b and c are real numbers such that a ≠ 0 and x is a variable.
Since the degree of the above-written equation is two; it will have two roots or solutions. The roots of polynomials are the values of x which satisfy the equation. There are several methods to find the roots of a quadratic equation. One of them is by completing the square. The methods of solving these quadratics have been introduced to the students in Class 10 and 11. Let us see the complete method here in this article.
Completing the Square for Quadratic Equation
We have our quadratic equation as;
ax2 + bx + c = 0
The roots of the quadratic equation can also be evaluated by using the quadratic formula. For simplification, let us take a=1. The equation hence becomes
x2 + bx + c = 0
If we wanted to represent a quadratic equation using geometry, one way would be representing the terms of the expression in the L.H.S. of the equation by using geometric figures such as squares, rectangles etc. If we take a square with the side equal to x units, its area would be equal to x2 square units. This area will hence represent the first term of the expression. Similarly, a rectangle with its two sides as x units and b units will have the area equal to bx square units. And let us take a square with area equal to c square units to represent the last term of the expression. In the figure below, we have the geometrical equivalent of the expression x2, bx and c.
Geometrical equivalent of x2,bx and c
This method is known as completing the square. Let us complete some squares. If we break the rectangle representing bx into two equal parts cutting vertically, we will have two figures with an area of each equal to b/2 x square units. The figures are arranged accordingly in the second figure, below. Now, we have
x2 + bx + c = 0
Rearranging the figures
But our square is not complete yet. To complete the square, one square of side b/2 units is needed. This final part of the main square can be taken from the square with the area c square units. Cutting it out and putting it at the place, it results in below figure.
Completing the Square
The square is finally complete. The area of the square is equal to
(x+b/2)2 square units
The remaining area is equal to
(c-b2/4) square units
All this time, we were rearranging the same figures that we had initially. It would hence be correct to say that
x2 + bx + c = (x+b/2)2 + (c-b2/4)
This method is known as completing the square method. We have achieved it geometrically. We know that x2 + bx + c = 0. So,
(x+b/2)2 + (c-b2/4) = 0
⇒ (x+b/2)2 = -(c-b2/4)
All the terms in the R.H.S. of the above equation are known. That’s why it is very easy to determine the roots. Let us look at some examples for better understanding.
Steps for Completing the square method
Suppose ax2 + bx + c = 0 is the given quadratic equation. Then follow the given steps to solve it by completing square method.
- Write the equation in the form, such that c is on the right side.
- If a is not equal to 1, then divide the complete equation by a, such that co-efficient of x2 is 1.
- Now add the square of half of the co-efficient of term-x, (b/2a)2, on both the sides.
- Factorize the left side of the equation as the square of the binomial term.
- Take the square root on both the sides
- Solve for variable x and find the roots.
Example 1: Find the roots of the quadratic equation x2 + 4x – 5 = 0 by the method of completing the square.
Given quadratic equation is:
x2 + 4x – 5 = 0
Comparing the equation with the standard form,
b = 4, c = -5
(x+b/2)2 = -(c-b2/4)
So, (x+4/2)2 = -(-5-42/4)
(x + 2)2 = 5 + 4
⇒ (x+2)2 = 9
⇒ (x+2) = ±√9
⇒ (x + 2) = ± 3
⇒ x + 2 = 3, x + 2 = -3
⇒ x = 1 , -5
Example 2: Find the roots of the quadratic equation 3x2 – 5x + 2 = 0 by the method of completing the square.
Given quadratic equation is:
3x2 – 5x + 2 = 0
The given equation is not in the form to which we apply the method of completing squares, i.e. the coefficient of x2 is not 1. To make it 1, we need to divide the whole equation with 3.
x2 – 5/3 x + 2/3 = 0
Comparing with the standard form,
b = -5/3; c = ⅔
c – b2/4 = ⅔ – [(5/3)2/4] = ⅔ – 25/36 = -1/36
⇒ (x – 5/6)2 = 1/36
⇒ (x – 5/6)= ± √(1/36)
⇒ x – 5/6 = ±1/6
⇒ x = 1, -2/3
The below video will help you visualize the concepts of quadratic equations.
To solve more problems on the topic, download BYJU’S – The Learning App from Google Play Store.