Completing the Square: Tips for Solving Equations with Coefficients

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SUMMARY

The discussion focuses on the method of completing the square for quadratic equations that include a leading coefficient. The key technique involves factoring out the leading coefficient from all terms before applying the completing the square formula, specifically using the identity (x-a)² = x² + 2ax + a². An example provided illustrates this process with the equation 4(y² − y + 1/4) − 3, leading to the final form 4(y − 1/2)² − 4.

PREREQUISITES
  • Understanding of quadratic equations and their standard form.
  • Familiarity with the concept of completing the square.
  • Basic algebraic manipulation skills, including factoring.
  • Knowledge of the identity (x-a)² = x² + 2ax + a².
NEXT STEPS
  • Practice completing the square with various leading coefficients.
  • Explore the derivation of the quadratic formula from completing the square.
  • Learn about the applications of completing the square in solving real-world problems.
  • Investigate the relationship between completing the square and graphing quadratic functions.
USEFUL FOR

Students studying algebra, mathematics educators teaching quadratic equations, and anyone looking to enhance their problem-solving skills in algebraic contexts.

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So far all the completing the square problems I've done do not have a coefficient in front of the highest power. This one has me stuck. :( What should I do to complete the square?
 
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Factor the leading coefficient out of all the terms and then complete the square.
 
(x-a)2= x2+ 2ax+ a2. After you have factored out that leading 4, what is "a"?
 
So would it be something like this?

4(y2 − y +1/4) − 3 − 1
4(y− 1/2)2 − 4
 
Last edited:
It would be a lot like that.
 

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