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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  =  ±Square root of b.
 
         Therefore,
x + 3  =  ±Square root of 7
 
x  =  −3 ±Square root of 7.

That is, the solutions to

x2 + 6x + 2  =  0

are the conjugate pair,

−3 + Square root of 7,  −3 − Square root of 7.

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 − (Square root of 7)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  =  ±Square root of 3
 
x  =  1 ± Square root of 3

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

The quadratic formula

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:

Complete the square Complete the square Complete the square Complete the square

This is the quadratic formula.

End of the lessson

Next Topic:  Synthetic division by xa


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