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Solution 2.3:2d

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The equation can be written in normalized form (i.e. the coefficient in front of x² is 1) by dividing both sides by 4,

x2−7x+413=0.

Complete the square on the left-hand side,

x2−7x+413=x−272−272+413=x−272−449+413=x−272−436=x−272−9.

The equation can therefore be written as

x−272−9=0 

and taking the square root gives the solutions as

  • x−27=9=3  i.e. x=27+3=213
  • x−27=−9=−3  i.e. x=27−3=21

As an extra check, we substitute x = 1/2 and x = 13/2 into the equation:

  • x = 1/2:  LHS=4212−2821+13=441−14+13=RHS, 
  • x = 13/2:  LHS=42132−28213+13=44169−1413+13=RHS. 
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