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Quadratic equations:  Section 2

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Completing the square

The quadratic formula

The discriminant

Proof of the quadratic formula

IN LESSON 18, we saw a technique called completing the square.  We will now see how to apply it to solving a quadratic equation.

Completing the square

If we try to solve this quadratic equation by factoring,

x² + 6x + 2 = 0
 
 we cannot.  Therefore, we complete the square.  That means to make the quadratic into 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 − .

For a method of checking these roots, see Lesson 10 of Topics in Precalculus,

In Lesson 18, there are examples and problems in which the coefficient of x is odd.  Also, some of the quadratics below have complex roots, and some involve simplifying radicals.

Problem 6.   Solve each quadratic equation by completing the square.

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   a)   x² − 2x − 2  =  0   b)   x² + 4x − 6  =  0
 
  x² − 2x  =  2   x² + 4x  =  6
 
  x² − 2x + 1  =  2 + 1   x² + 4x + 4  =  6 + 4
 
  (x − 1)²  =  3   (x + 2)²  =  10
 
  x − 1  =  ±

  x + 2  =  ±
 
  x  =  1 ±   x  =  −2 ±
   c)   x² − 4x + 13  =  0   d)   x² + 6x + 29  =  0
 
  x² − 4x  =  −13   x² + 6x  =  −29
 
  x² − 4x + 4  =  −13 + 4   x² + 6x + 9  =  −29+ 9
 
  (x − 2)²  =  −9   (x + 3)²  =  −20
 
  x − 2  =  ±3i

  x + 3  =  ±
 
  x  =  2 ± 3i   x  =  −3 ± 2i
   e)   x² − 5x − 5  =  0   f)   x² + 3x + 1  =  0
 
  x² − 5x  =  5   x² + 3x  =  −1
 
  x² − 5x + 25/4  =  5 + 25/4   x² + 3x9/4  =  −1 + 9/4
 
  (x5/2  =  5 + 25/4   (x + 3/2  =  − 1 + 9/4
 
  x5/2  =  ±

  x + 3/2  =  ±
 
  x  =    x  = 

The quadratic formula

Here is the quadratic formula -- which is proved by completing the square!  In other words, the quadratic formula completes the square for us.

Theorem.    If

ax² + bx + c = 0,

Theorem.   then

We will prove this below.

Example 4.   Use the quadratic formula to solve this quadratic equation:

3x² + 5x − 8  =  0

Solution.   We have:   a  =  3,  b  =  5,  c  =  −8.

Therefore, according to the formula:

x =
 
  =
 
  =
 
  =

That is,

x = −5 + 11
     6
 or   −5 − 11
     6
x     =   6
6
   or    −16
  6
x     =   1    or   − 8
3
.

These are the two roots.  And they are rational.  This tells us that we could have solved the equation by factoring, which is always the simplest method.

3x² + 5x − 8   =  (3x + 8)(x − 1)
 
x  =  8
3
  or  1.

Problem 7.   Use the quadratic formula to find the roots of each quadratic.

a)   x² − 5x + 5

a  =  1,  b  = −5,  c  =  5.

x = = =

b)  2x² − 8x + 5

a  =  2,  b  = −8,  c  =  5.

x =
 
  = = = = 2 + ½

c)  5x² − 2x + 2

a  =  5,  b  = −2,  c  =  2.

x =
 
  = = = 2 ± 6i
   10
= 1 ± 3i
   5

The discriminant

The radicand  b² − 4ac  is called the discriminant.  If the discriminant is

a)   Positive: The roots are real and conjugate.
 
b)   Negative:   The roots are complex and conjugate.
 
c)   Zero: The roots are rational and equal -- i.e. a double root.

Proof of the quadratic formula

To prove the quadratic formula, we 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:

on multiplying both c and a by 4a, thus making the denominators the same (Lesson 23),

This is the quadratic formula.  

Section 3:  The graph of  y = A quadratic

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