**Solving Quadratic Equations by Completing the Square Method :**

Here we are going to see how we use the method completing the square to solve a quadratic equation.

**Step 1 :**

Write the quadratic equation in general form ax^{2} +bx +c = 0 .

**Step 2 :**

Divide both sides of the equation by the coefficient of x2 if it is not 1.

**Step 3 :**

Shift the constant term to the right hand side.

**Step 4 :**

Add the square of one-half of the coefficient of x to both sides.

**Step 5 :**

Write the left hand side as a square and simplify the right hand side.

**Step 6 :**

Take the square root on both sides and solve for x.

**Question 1 :**

Solve the following quadratic equations by completing the square method

(i) 9x^{2} −12x + 4 = 0

**Solution :**

**Step 1 :**

Since the coefficient of x^{2} is 9 not 1, we have to divide the entire equation by 9.

x^{2} - (4/3)x + (4/9) = 0

**Step 2 :**

Shift the constant term to R.H.S

x^{2} - (4/3)x = -4/9

**Step 3 :**

Half of coefficient of x is -2/3. We have to square this and add it on both sides.

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

[x - (2/3)]^{2} = (-4/9) + (4/9)

[x - (2/3)]^{2} = 0

[x - (2/3)]^{2} = 0

x - (2/3) = 0 and x - (2/3) = 0

x = 2/3 and x = 2/3.

Hence the solutions are 2/3 and 2/3.

(ii) (5x + 7)/(x - 1) = 3x + 2

**Solution :**

5x + 7 = (3x + 2)(x - 1)

5x + 7 = 3x^{2} - 3x + 2x - 2

3x^{2} - x - 5x - 2 - 7 = 0

3x^{2} - 6x - 9 = 0

Divide the entire equation by 3.

x^{2} - 2x - 3 = 0

x^{2} - 2x = 3

Half of coefficient of x is -1. Now we are going to add the square of -1 on both sides.So, we get

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

x^{2} - 2x + 1 = 3 + 1

(x - 1)^{2} = 4

(x - 1) = √4

(x - 1) = ± 2

x - 1 = 2 and x - 1 = -2

x = 2 + 1 and x = -2 + 1

x = 3 and x = -1

Hence the solutions are 3 and -1.

After having gone through the stuff given above, we hope that the students would have understood, "Solving Quadratic Equations by Completing the Square Method".

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