**Completing the Square Method Examples with Answers :**

In this section, we will learn how to solve quadratic equations using completing the square method.

**Step 1 :**

In the given quadratic equation ax^{2} + bx + c = 0, divide the complete equation by a (coefficient of x^{2}).

If the coefficient of x^{2} is 1 (a = 1), the above process is not required.

**Step 2 :**

Move the number term (constant) to the right side of the equation.

**Step 3 :**

In the result of step 2, write the "x" term as a multiple of 2.

Examples :

6x should be written as 2(3)(x).

5x should be written as 2(x)(5/2).

**Step 4 :**

The result of step 3 will be in the form of

x^{2} + 2(x)y = k

**Step 4 :**

Now add y^{2} to each side to complete the square on the left side of the equation.

Then,

x^{2} + 2(x)y + y^{2} = k + y^{2}

**Step 5 :**

In the result of step 4, if we use the algebraic identity

(a + b)^{2} = a^{2} + 2ab + b^{2}

on the left side of the equation, we get

(x + y)^{2} = k + y^{2}

**Step 6 :**

Solve (x + y)^{2} = k + y^{2 }for x by taking square root on both sides.

**Example 1 :**

Solve by completing the square method

x^{2} – (√3 + 1)x + √3 = 0

**Solution :**

x^{2} – (√3 + 1)x + √3 = 0

x^{2} – (√3 + 1)x = -√3

x^{2} – (√3 + 1)x + [(√3 + 1)/2]^{2 }= -√3 + [(√3 + 1)/2]^{2}

[x - (√3 + 1)/2]^{2 }= -√3 + [(√3 + 1)^{2} / 4]

[x - (√3 + 1)/2]^{2 }= (-4√3 + √3^{2} + 2√3 + 1)/4

[x - (√3 + 1)/2]^{2 }= (√3^{2} - 2√3 + 1)/4

[x - (√3 + 1)/2]^{2 }= [ (√3 - 1)/2 ]^{2}

[x - (√3 + 1)/2]^{2 }= [ (√3 - 1)/2 ]^{2}

[x - (√3 + 1)/2]^{ }= ± (√3 - 1)/2

x - (√3+1)/2 = (√3-1)/2 x = (√3-1)/2 + (√3+1)/2 x = 2√3/2 x = √3 |
x - (√3+1)/2 = -(√3-1)/2 x = -(√3-1)/2 + (√3+1)/2 x = 2/2 x = 1 |

Hence the solution is { 1, √3 }.

**Question 6 :**

Solve by completing the square method

(5x + 7)/(x – 1) = 3x + 2

**Solution :**

(5x + 7) = (3x + 2)(x – 1)

5x + 7 = 3x² – 3x + 2x – 2

3x² – 3x + 2x – 2 – 5x – 7 = 0

3x² – 6x – 9 = 0

Divide the entire equation by 3, we get

x² – 2x – 3 = 0

x² – 3x + x – 3 = 0

x (x – 3) + 1 (x – 3) = 0

(x – 3)(x + 1) = 0

x - 3 = 0 x = 3 |
x + 1 = 0 x = -1 |

Hence the solution is { -1, 3 }.

After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations using the completing the square.

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