**Solving quadratic equations by completing the square examples :**

To make any quadratic expression a perfect square, a method called completing the square may be used.

To complete the square for a quadratic expression of the form x^{2} + bx (or) x^{2} - bx, we can follow the steps below.

Step 1 : Find 1/2 of b, the coefficient of x.

Step 2 : Square the result of Step 1.

Step 3 : Add the result of Step 2 to x^{2} + bx, the original expression

Let us look into some example problems to understand the above concept.

**Example 1 :**

Solve a^{2} - 14a + 3 = -10 by completing the square.

**Solution :**

In order to use the method "completing the square", the given question must be in the form x^{2} + bx.

So we need to subtract 3 on both sides,

a^{2} - 14a + 3 - 3 = -10 - 3

a^{2} - 14a = -13

**Step 1 :**

Coefficient of 1/2 of a. That is 14/2 = 7

**Step 2 :**

Square the result of step 1. That is 7^{2} = 49

**Step 3 :**

a^{2} - 14a + 7^{2} = -13 + 7^{2}

a^{2} - 2 **⋅ **a **⋅** 7 + 7^{2} = -13 + 7^{2}

On the left we have the form a^{2}- 2ab +b^{2}.Instead of this we can write (a -b)^{2}

(a - 7)^{2} = -13 + 49

(a + 7)^{2} = 36

Taking square roots of both sides,

√(a + 7)^{2} = √36

(a + 7) = ±6

a - 7 = 6 Add 7 on both sides a - 7 + 7 = 6 + 7 a = 13 |
a - 7 = -6 Add 7 on both sides a - 7 + 7 = -6 + 7 a = 1 |

Hence the solution is {1, 13}

Let us see the next example on "Solving quadratic equations by completing the square examples".

**Example 2 :**

Solve x^{2} - 6x + 9 = 25 by completing the square.

**Solution :**

In order to use the method "completing the square", the given question must be in the form x^{2} + bx (or) x^{2} - bx.

To convert the given question in the above form, we need to subtract 9 on both sides,

x^{2} - 6x + 9 - 9 = 25 - 9

x^{2} - 6x = 16

**Step 1 :**

Coefficient of 1/2 of x. That is 6/2 = 3

**Step 2 :**

Square the result of step 1. That is 3^{2} = 9

**Step 3 :**

x^{2} - 6x + 3^{2 } = 16 + 3^{2}

x^{2} - 2 **⋅ x** **⋅** 3 + 3^{2} = 16 + 9

On the left we have the form a^{2}- 2ab +b^{2}.Instead of this we can write (a -b)^{2}

(x - 3)^{2} = 25

Taking square roots of both sides,

√(x - 3)^{2} = √25

(x - 3) = ±5

x - 3 = 5 Add 3 on both sides x - 3 + 3 = 5 + 3 x = 8 |
x - 3 = -5 Add 3 on both sides x - 3 + 3 = -5 + 3 x = -2 |

Hence the solution is {8, -2}

Let us see the next example on "Solving quadratic equations by completing the square examples".

**Example 3 :**

Solve x^{2} + 14x + 49 = 20 by completing the square.

**Solution :**

In order to use the method "completing the square", the given question must be in the form x^{2} + bx (or) x^{2} - bx.

To convert the given question in the above form, we need to subtract 49 on both sides,

x^{2} + 14x + 49 - 49 = 20 - 49

x^{2} + 14x = -29

**Step 1 :**

Coefficient of 1/2 of x. That is 14/2 = 7

**Step 2 :**

Square the result of step 1. That is 7^{2} = 49

**Step 3 :**

x^{2} + 14x + 7^{2 }= -29 + 7^{2}

x^{2} + 2 **⋅ x** **⋅** 7 + 7^{2 }= -29 + 49

(x + 7)^{2} ^{ }= 20

On the left we have the form a^{2} + 2ab +b^{2}.Instead of this we can write (a +b)^{2}

(x + 7)^{2} ^{ }= 20

Taking square roots of both sides,

√(x + 7)^{2} = √20

(x + 7) = ±√20

x + 7 = √20 Subtract 7 on both sides x + 7 - 7 = √20 - 7 x = 2√5 - 7 |
x + 7 = -√20 Subtract 7 on both sides x + 7 - 7 = -√20 - 7 x = -2√5 - 7 |

Hence the solution is {2√5 - 7, -2√5 - 7}

Let us see the next example on "Solving quadratic equations by completing the square examples".

**Example 4 :**

Find the value of c that makes each trinomial a perfect square.

a^{2} - 12a + c

**Solution :**

= a^{2} - 12a + c

= a^{2} - 2 **⋅** a **⋅** 6 + c

Instead of "c" if we write 6^{2}, the given polynomial will become a perfect square.

= a^{2} - 2 **⋅** a **⋅** 6 + 6^{2}

Since the above polynomial is in the form a^{2} - 2ab + b^{2}, we can write it as (a - b)^{2}

= (a + 6)^{2}

Hence the value of c is 6^{2}, that is 36.

We hope that the students would have understood the stuff and example problems explained on "Solving quadratic equations by completing the square examples".

Apart from the stuff and example problems explained above, if you want to know more about "Solving quadratic equations by completing the square examples", please click here

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