HOW TO SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE

If possible, solve for x using ‘completing the square’ :

Example 1 :

x² + 2x  =  5

Solution :

The coefficient of x² is 1, so we dont have to divide the entire equation.

x² + 2x  =  5

Half of the coefficient of x(2/2) is 1.

x² + 2x + 1²  =  5 + 1²

x² + 2x + 1  =  6

(x + 1)²  =  6

Taking square roots on both sides, we get

(x + 1)  =  ±√6

So, solution are √6-1 and -√6-1.

Example 2 :

x² - 2x  =  - 7

Solution :

x² - 2x  =  - 7

Coefficient of x² is 1. So dividing the entire equation is not required.

Coefficient of x is -2, half of it  =  -1

x² - 2x + (-1)²  =  - 7 + (-1)²

x² - 2x + 1  =  - 7 + 1

x² - 2x + 1  =  - 6

(x - 1)²  =  - 6

So, the system has no real solution.

Example 3 :

x² - 6x  =  - 3

Solution :

x² - 6x  =  - 3

Coefficient of x² is 1. So dividing the entire equation is not required.

Coefficient of x is -6, half of it  =  -3

x² - 6x + (-3)²  =  - 3 + (-3)²

x² - 6x + 9  =  - 3 + 9

x² - 6x + 9  =  6

(x - 3)²  =  6

(x - 3)  =  ±√6

So, the solutions are √6+3 and -√6+3.

Example 4 :

Solve by completing the square method 

2x² + 5x - 3  =  0

Solution :

2x² + 5x - 3  =  0

2x² + 5x  =  3/2

Divide the equation by 2.

x² + (5/2)x  =  (3/2)

Half of coefficient of x is 5/4.

By adding (5/4)² on both sides, we get 

x² + (5/2)x + (5/4)²  =  (3/2) + (5/4)²

x² + (5/2)x + (5/4)²  =  (3/2) + (5/4)² 

[ x + (5/4) ]²  =  (3/2) + (25/16)

[ x + (5/4) ]²  =  49/16

[ x + (5/4) ]²  =  √(49/16)

x + (5/4) = ± 7/4

So, the solutions area 1/2 and -3.

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