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Solved Examples for Completing the Square

Example 1:   x2 + 5x + 2 = 0

Solution:
Steps 1, 2 and 3:
The given equation is already in the standard form, so there is no need to convert it to the standard form. Further, the coefficient of x-squared is 1, so on dividing the equation by it the equation will remain unchanged; So there is no need to divide the equation by the coefficient of x-squared. Move the constant term to RHS,

x2 + 5x = -2
Step 4:
  • Take the coefficient of 'x': 5
  • Divide it by 2: 5/2
  • Square it: (5/2)2
  • Add it to both sides of the equation:
Example 1 Step 4

Step 5:
Comparing the LHS with a2 + 2ab + b2, we obtain 'a' = x and 'b' = (5/2). Now since the LHS is in the form of a2 + 2ab + b2, therefore it can be rewritten as (a + b)2, because of the mathematical property (a + b)2 = a2 + 2ab + b2. Thus,

Example 1 Step 5

Step 6:
Now that x-squared has been completely factored from the equation, it can be solved for 'x'. Take square roots of both sides in order to remove the square on LHS,

Example 1 Step 6 A

The square root on the LHS cancels out the exponent of 2,

Example 1 Step 6 B

Simplifyt he RHS,

Example 1 Step 6 C

The square root of a number can be positive or negative, so put a 'plus below minus' sign before the RHS, and move fraction 5/2 to RHS,

Example 1 Step 6 D

Solve further,

Example 1 Step 6 E

Answer: Answer of Example 1


Example 2:   2x2 + 4x = -2

Solution:
Step 1: Convert to Standard Form

Example 2 Step 1

Step 2: Divide by coefficient of x-squared:

Example 2 Step 2

Step 3: Move constant term to RHS:

Example 2 Step 3

Step 4: Take coefficient of 'x', divide it by 2, then square it. Add the result to LHS and RHS,

Example 2 Step 4

Step 5: Compare the LHS with a2 + 2ab + b2 and apply the formula (a + b)2 = a2 + 2ab + b2

Example 2 Step 5

Step 6: Solve for 'x':

Example 2 Step 6

Answer: x = -1


Example 3:   6p2 = -11p - 10

Solution:
Step 1: Convert to Standard Form

6p2 + 11p + 10 = 0
Step 2: Divide by coefficient of p-squared:

Example 3 Step 2

Step 3: Move constant term to RHS:

Example 3 Step 3

Step 4: Take coefficient of 'p', divide it by 2, then square it. Add the result to LHS and RHS,

Example 3 Step 4

Step 5: Compare the LHS with a2 + 2ab + b2 and apply the formula (a + b)2 = a2 + 2ab + b2

Example 3 Step 5

Step 6: Solve for 'p' (by taking square roots on both sides):

Example 3 Step 6

Answer: Answer

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