A polynomial equation with degree equal to two is known as quadratic equation.‘Quad’ means four but ‘Quadratic’ means ‘to make square’. A quadratic equation in its standard form is represented as:

\( ax^2~+~bx~+~c\)

Since,the degree of the above written equation is two; it will have two roots or solutions. The roots of an equation 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 method.

## Completing the Square: fundamentals

We have our quadratic equation as

\( ax^2~+~bx~+~c\)

For simplification, let us take a=1. The equation hence becomes

\(x^2~+~bx~+~c\)

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 \(x^2\)

Figure 1: Geometrical equivalent of \(x^2\)

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 area of each equal to \( \frac b2 \)

\(x^2~+~bx~ and~ c \)

Figure 2: Rearranging the figures

But our square is not complete yet. To complete the square, one square of side \( \frac b2 \)

Figure 3: Completing the Square

The square is finally complete. The area of the square is equal to

\( \left( x~+~\frac b2 \right)^2 \)

The remaining area is equal to

\( \left( c~-~\frac {b^2}{4} \right) \)

All this time, we were rearranging the same figures that we had initially. It would hence be correct to say that

\( x^2~+~bx~+~c\)

This method is known as completing the square method. We have achieved it geometrically. We know that \(x^2~+~bx~+~c\)

\( \left( x~+~\frac b2 \right)^2~+~\left(c~-~\frac{b^2}{4}\right) \)

⇒ \( \left( x~+~\frac b2 \right)^2 \)

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.

*Example 1:* \( x^2~+~4x~-~5\)

So,\( \left( x~+~\frac 42 \right)^2 \)

⇒ \((x~ +~ 2)^2\)

⇒ \( (x~+~2)\)

⇒ (x + 2 ) = ± 3

⇒ x = 1 , -5

*Example 2: *\( 3x^2~-~5x~+~2\)

The given equation is not in the form to which we apply method of completing squares, i.e. coefficient of \( x^2\)

\( x^2~-~\frac 53 x~+~\frac 23\)

b = \( – \frac 53\)

So, \( \left( x~+~\frac{\left( – \frac 53 \right)}{2} \right)^2 \)

⇒ \( \left( x~-~\frac56 \right)^2 \)

⇒ \( \left( x~-~\frac56 \right) \)

⇒ \( \left( x~-~\frac 56 \right) \)

⇒ \( x \)

The below video will help you visualize the concepts of quadratic equations

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To solve more problems on the topic, download Byju’s – The Learning App from Google Play Store.