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11

COMPLETING THE SQUARE

IF WE TRY TO SOLVE this quadratic equation by factoring

x² + 6x + 2  =  0

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  a² + 2ab + b²  =  (a + b)².

The technique is valid only when 1 is the coefficient of x².

1)  Transpose the constant term to the right:

x² + 6x  = −2

2)  Add a square number to both sides.  Add the square of half the coefficient of x.  In this case, add the square of 3:

x² + 6x + 9  =  −2 + 9

The left-hand side is now the perfect square of  (x + 3).

(x + 3)²  =  7.

3 is half of the coefficient 6.

This equation has the form

a²  =  b
  which implies
a  =  ±.
 
         Therefore,
x + 3  =  ±
 
x  =  −3 ±.

That is, the solutions to

x² + 6x + 2  =  0

are the conjugate pair,

−3 + ,  −3 − .

We can check this.  The sum of those roots is −6, which is the negative of the coefficient of x.  And the product of the roots is

(−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.

x² − 2x − 2 = 0

To see the solution, pass your mouse over the colored area.
To cover the solution again, click "Refresh" ("Reload").

x² − 2x  =  2
 
x² − 2x + 1  =  2 + 1
 
(x − 1)²  =  3
 
x − 1  =  ±

 
x  =  1 ±

See Lesson 37 of Algebra, Problems 6 and 7.

Before considering the quadratic formula, note that half of any

 number b is   b
2
.   Half of  5 is  5
2
.   Half of   p
q
 is    p
2q
.

(Lesson 27 of Arithmetic, Question 4.)

The quadratic formula

Theorem.   If

ax² + bx + c = 0,

Theorem.   then

To prove this, we will complete the square.  But to do that, the coefficient of x² must be 1.  Therefore, we will divide both sides of the original equation by a:

This is the quadratic formula.


Next Topic:  Synthetic division by xa


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