"Solving quadratic equations by completing the square" is one of the methods being applied to solve any kind of quadratic equation.

First let us understand the method and look at some example problems for practice.

(i) First we have to check whether the coefficient of x² is 1 or not. If yes we can follow the second step. Otherwise we have to divide the entire equation by the coefficient of x².

(ii) Bring the constant term which we find on the left side to the right side.

(iii) We have to add the square of half of the coefficient of "x" on both sides.

(iv) Now the three terms on the left side will be in the form of a² + 2 a b + b² (or) a² - 2 ab + b².

(v) Then, we can write (a + b)² for a² + 2 a b + b² and (a- b)² for a² - 2 a b + b². Then we have to solve for x by simplification.

(1) Solve the following quadratic equations by completing the square.

**Problem 1:**

Solve the quadratic equation x² + 6 x - 7 = 0 by completing the square method

**Solution:**

(x + 3)² = 16

x + 3 = √ 16

x + 3 = ± 4

x + 3 = 4 x + 3 = - 4

x = 4 - 3 x = - 4 - 3

x = 1 x = - 7

**Problem 2:**

Solve the quadratic equation x² + 3 x + 1 = 0 by completing the square method

**Solution:**

x² + 3 x + 1 = 0

x² + 3 x = -1

x²+ 2 **x** x **x **(3/2) + (3/2)² = - 1 + (3/2)²

[x + (3/2)]² = - -1 + (9/4)

(x + (3/2))² = (9/4) - 1

(x + (3/2))² = (5/4)

(x + (3/2)) = √(5/4)

x + (3/2) = ± (√5/2)

x + (3/2) = (√5/2) x + (3/2) = -(√5/2)

x = (√5/2) - (3/2) x = -(√5/2) - (3/2)

x = (√5 - 3)/2 x = (-√5 - 3)/2

**Problem 3:**

Solve the quadratic equation 2 x² + 5 x - 3 = 0 by completing the square method

**Solution:**

2 x² + 5 x - 3 = 0

divide the whole equation by 2

x² + (5/2) x - (3/2) = 0

x² + (5/2) x = (3/2)

x² + 2 (5/2) x = (3/2)

x² + 2 x (5/2) + (5/2)² = (5/2)²+ (3/2)

(x + (5/2))² - (25/4)- (3/2) = 0

(x + (5/2))² = (25/4) + (3/2)

(x + (5/2))² = (25 + 6)/4

(x + (5/2))² = 31/4

x + (5/2) = √(31/4)

x + (5/2) = ± √31/2

x + (5/2) = √31/2 x + (5/2) = -√31/2

x = (√31/2) - (5/2) x = (-√31/2) - (5/2)

x = (√31 - 5)/2 x = (- √31-5)/2

**Problem 4:**

Solve the quadratic equation 4 x² + 4 b x - (a² - b²) = 0 by completing the square method

**Solution:**

4 x² + 4 b x - (a² - b²) = 0

dividing the whole equation by 4,we get

x² + b x - (a² - b²)/4= 0

x = (a - b)/2 or x = (-a -b)/2

**Problem 5:**

Solve the quadratic equation x² - ( √3 + 1) x + √3 = 0 by completing the square method

**Solution:**

x² - ( √3 + 1) x + √3 = 0

x = -1 , -√3

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

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

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