You might be knowing the quadratic equations. Any equation in the form of ax^{2}+bx+c = 0 is termed as quadratic. Where a, b, and c are real numbers such that a ≠ 0 and x is a variable. As it is a second-degree equation, it has got two roots. There are many ways to find these roots. Method of completing the square is the most commonly used.

The method of completing the squares is beneficial for solving, deriving, and graphing quadratic equations. You can also use it in evaluating integrals in calculus and Laplace transforms. Let us learn in detail about this method of completing the squares.

## Formula of Completing the Square Method

Solving quadratic equations that cannot be factored in involves completing the square. In this method, the given quadratic equation is manipulated to get the perfect square trinomial on the left side of the equation.

The formula is derived by manipulating the quadratic equation of the form ax^{2 }+ bx + c = 0 as given below.

To the right of the equation, isolate the term c

ax^{2 }+ bx = -c

Divide each term by the coefficient of x^{2}. That is a.

x^{2 }+ bx/a = -c/a

Add the square of half of the coefficient of x, (b/2a)^{2}, on both sides.

x ^{2 }+ bx/a + (b/2a)^{2 }= – c/a + (b/2a)^{2}

Write the LHS term as a perfect square.

(x + b/2a)^{2}= (-4ac+b^{2})/4a^{2}

Taking square root on both sides

(x + b/2a) = ±√ (-4ac+b^{2})/2a

x = – b/2a ±√ (b^{2}– 4ac)/2a

x = -bb2-4ac2a………. (This is the required quadratic formula)

## Steps to Solve a Quadratic Equation using Completing the Square Method

To solve the quadratic equation of the standard form ax^{2}+bx+c = 0 by the method of completing squares follow the below steps.

Write the given equation in the standard form. Then take c to the RHS side of the equation.

Make the coefficient of x^{2} equal to 1 by dividing the entire equation by the given coefficient.

Add the square of half of the coefficient of x, (b/2a)^{2}, on both sides.

Write the LHS term as a perfect square.

Find the square root of the two sides

Calculate the roots of the variable x.

Let me explain these steps in detail with this example

**Example: **Find the roots of the quadratic equation 2x^{2} + 5x – 3 = 0 by the method of completing the square.

**Solution:** Given: quadratic equation is 2x^{2} + 5x – 3 = 0. It is in the standard form.

Take c term to the RHS side of the equation

∴ 2x^{2} + 5x = 3

Make the coefficient of x^{2} equal to 1 by dividing the entire equation by the given coefficient.

2x^{2} + 5x = 3 2 on both sides

x^{2} + 5/2x = 3/2

Add the square of half of the coefficient of x, (b/2a)^{2}, on both sides. Here b = 5 and a = 2

x^{2} + 5/2x + (5/4)^{2} = 3/2 + (5/4)^{2}

The LHS is in the form of (a + b)^{2}. Hence x^{2} + 5/2x + (5/4)^{2} = (x + 5/4)^{2}

(x + 5/4)^{2} = 3/2 + (25/16) = (24 + 25)/16 = 49/16

Find the square root of the two sides

(x + 5/4)^{2}= 49/16

x + 5/4 = 7/4

x = 7/4 – 5/4 = 2/4 = 1/2

Hence the root of x is ½.

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