11 ## COMPLETING THE SQUAREIF WE TRY TO SOLVE this quadratic equation by factoring
we cannot. Therefore, we use a technique called completing the square. That means to make the quadratic into a perfect square trinomial, i.e. the form The technique is valid only when 1 is the coefficient of 1) Transpose the constant term to the right:
2) Add a square number to both sides. Add the square of
The left-hand side is now the perfect square of ( ( 3 is This equation has the form
That is, the solutions to
are the conjugate pair, −3 + , −3 − . We can check this. The sum of those roots is −6, which is the (−3)² − ()² = 9 − 7 = 2, which is the constant term. Thus both conditions on the roots are satisfied. These are the two roots of the quadratic. Problem. Solve this quadratic equation by completing the square.
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See Lesson 37 of Algebra, Problems 6 and 7. Before considering the quadratic formula, note that half of any
(Lesson 27 of Arithmetic, Question 4.) The quadratic formula
Theorem.
Theorem. To prove this, we will complete the square. But to do that, the coefficient of This is the quadratic formula. Next Topic: Synthetic division by Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |