**If the given quadratic equation is not a perfect square, then we need to make that as a perfect square. To make any quadratic expression as a perfect square, we need to use the method of completing the square as listed:-**

**1) Check for the following for the given quadratic equation,**

*ax*^{2}+*bx*+*c*= 0.**(a) If the given quadratic equation has the coefficient of**

*x*^{2}term as 1, then proceed to step (c).**(b) If the coefficient of**

*x*^{2}is not 1, then make it 1 by dividing both the sides of the equation by the coefficient.**(c) Isolate the**

*x*and*x*^{2}terms on one side and the constants on the other side of the equation.**2) Divide the co-efficient of**

*x*by 2 (That is^{b}/_{2}).**3) Find the square of**

^{b}/_{2}(That is (^{b}/_{2})^{2}).**4) Add the resultant number (**

^{b}/_{2})^{2}to both sides of the given equation and simplify it.**Let us solve a few quadratic equations by using completing the square method.**

**Example 1:**

**Solve**

*x*^{2}+ 4*x*+ 3 = 0 by completing the square.**Solution:**

**Step 1:**

**The given quadratic equation is**

*x*^{2}+ 4*x*+ 3 = 0. To solve by completing the square rewrite the given equation as a perfect square.**Here, the coefficient of**

*x*^{2}is 1.**So, isolate the**

*x*and*x*^{2}terms to one side and the constant to the other side by subtracting 3 from both sides of the quadratic equation.

*x*^{2}+ 4*x*+ 3 - 3 = 0 - 3**=>**

*x*^{2 }+ 4*x*= -3**Step 2:**

**Now, divide the coefficient of**

*x*by 2.**=> 4 ÷ 2 = 2**

**Step 3:**

**Find the square of 2.**

**i.e., 2**

^{2}= 4**Step 4:**

**Now, add 4 on both the sides of the equation**

*x*^{2 }+ 4*x*= -3 and simplify.

*x*^{2}+ 4*x*+ 4 = -3 + 4**=>**

*x*^{2}+ 4*x*+ 4 = 1**Since,**

*x*^{2}+ 4*x*+ 4 = (*x*+ 2)^{2}_{, }the equation becomes**(**

*x*+ 2)^{2}= 1**Step 5:**

**Take square roots on both sides and simplify.**

**=>**

*x*+ 2 = 1 or*x*+ 2 = -1**=>**

*x*= 1 - 2 or*x*= -1 - 2**=>**

*x*= -1 or*x*= -3**So, the solutions are -1 and -3.**

**Step 6:**

**Check:**

**We can also check the solution by substituting it in the quadratic equation.**

**If**

*x*= -1, then

*x*^{2}+ 4*x*+ 3 = (-1)^{2}+ 4(-1) + 3**= 1 - 4 + 3**

**= 4 - 4 = 0**

**This is true.**

**If**

*x*= -3, then

*x*^{2}+ 4*x*+ 3 = (-3)^{2}+ 4(-3) + 3**= 9 - 12 + 3**

**= 12 - 12 = 0**

**This is true.**

**Step 7:**

**So, the solution set of the given quadratic equation is {-1, -3}.**

**Example 2:**

**A rectangular park is 10 m wide and 12 m in length. A pathway is constructed for pedestrians around the park. The area of the park including the pathway is 255 square meters. Find the width of the pathway.**

**Solution:**

**Step 1:**

**Length of the rectangular park = 12 m**

**Width of the rectangular park = 10 m**

**Let the width of the pedestrian path be**

*x*meters.

**Step 2:**

**Area of the rectangular park = Length * Width**

**= 12 m × 10 m = 120 square meters**

**Step 3:**

**Length of the park after constructing the pedestrian path = 12 +**

*x*+*x*= 12 + 2*x***New length = 12 + 2**

*x*meters**Width after constructing the pedestrian path = 10 +**

*x*+*x*= 10 + 2*x***New width = 10 + 2**

*x*meters**Area of the park including the path = 255 square meters**

**New length * New width = 255 square meters**

**(12 + 2**

*x*)(10 + 2*x*) = 255**120 + 24**

*x*+ 20*x*+ 4*x*^{2 }= 255**120 + 44**

*x*+ 4*x*^{2 }= 255**4**

*x*^{2}+ 44*x*+ 120 - 255 = 0**4**

*x*^{2}+ 44*x*- 135 = 0**Step 4:**

**Now, solve this quadratic equation by completing the square. The coefficient of**

*x*^{2}is not 1. So, divide the equation by 4 on both the sides.

*x*^{2}+ 11*x*–^{135}/_{4 }= 0**Isolate the**

*x*and*x*^{2}terms.**Add**

^{135}/_{4 }on both sides of the equation.

*x*^{2}+ 11*x*-^{135}/_{4 }+^{135}/_{4 }= 0 +^{135}/_{4}**=>**

*x*^{2}+ 11*x*=^{135}/_{4}**Step 5:**

**Now, the coefficient of**

*x*is 11. Divide it by 2 and square it. So, it becomes**(**

^{11}/_{2})^{2}=^{121}/_{4}**Step 6:**

**Now, add**

^{121}/_{4 }on both sides of the equation*x*^{2}+ 11*x*=^{135}/_{4}.

*x*^{2}+ 11*x*+^{121}/_{4}=^{135}/_{4}+^{121}/_{4}**=>**

*x*^{2}+ 11*x*+^{121}/_{4}=^{256}/_{4}**=>**

*x*^{2}+ 11*x*+^{121}/_{4}= 64**In the equation,**

*x*^{2}+ 11*x*+^{121}/_{4}is a perfect square. So, it the equation can be written as,**(**

*x*+^{11}/_{2})^{ 2}= 64**Take square roots of both the sides.**

**x +**

^{11}/_{2}= 8 or x -^{11}/_{2 }= -8**=>**

*x*= 8 -^{11}/_{2}or*x*= -8 -^{11}/_{2}**=>**

*x*=^{5}/_{2}or*x*= -^{27}/_{2}**=>**

*x*= 2.5 or*x*= -13.5**We are looking for the width of the pathway and the width (distance) cannot be negative.**

**So, ignore the negative number.**

**=>**

*x*= 2.5**Step 7:**

**Related Articles:**