The Quadratic Polynomial; The Quadratic Equation; The Quadratic Inequality

The quadratic polynomial; The quadratic equation; The quadratic inequality f(x) = ax2 + bx + c, a  0 ax2 + bx + c = 0, a  0 ax2 + bx + c > 0, a  0

Graph of the quadratic function  The

graph of f(x) = ax2 + bx + c is a parabola. a<0

a>0 axis of symmetry

minimu m point

maximum point

Completing the square  In

general, f(x) = ax2 + bx + c can be expressed in the form f(x) = a(x – p)2 + q by completing the square.

Completing the square  Example

1:  f(x) = x2 + 8x – 3 = x2 + 8x + 16 – 16 – 3 = (x + 4)2 – 19   (x + 4)2  0 for all real values of x,  f(x)  -19.  Or, the minimum point is at (- 4, -19)

When (x + 4)2 = 0, x = - 4.

Completing the square  





Example 2: g(x) = -2x2 + 6x + 5 = -2(x2 – 3x + 5/2) = -2(x2 – 3x + 9/4 - 9/4 + 5/2) = -2[(x – 3/2)2 + 1/4] = -2(x – 3/2)2 – ½  (x – 3/2)2  0 for all real values of x,  f(x)  – ½. Or, the maximum point is at (3/2, – ½)

When (x – 3/2)2 = 0, x = 3/2

The quadratic equation 

Derive the quadratic formula by completing the square:

ax 2  bx  c  0 

b  x    2a  

c  b x   x  0 a  a 2



2

x2   



2

b c  b  b  x     0 a a  2a   2a 

b  x  2a  

2

2

c  b    a  2a 

2

b 2  4ac  4a 2

b b 2  4ac x  2a 4a 2 b x  2a

b 2  4ac 2a

 b  b 2  4ac x 2a

Types of roots of a quadratic equation  The

nature of the roots of a quadratic equation depends on the value of the discriminant,  D = b2 - 4ac.

Types of roots of a quadratic equation  Case

1: When b2 – 4ac > 0, the roots are real and different. y

O

y

a>0

x

O

The graph intersects the x-axis at two different points.

a<0

x

Types of roots of a quadratic equation  Case

2: When b2 – 4ac = 0, the roots are real and equal. y

a>0

y

x

O The graph touches the x-axis.

O

a<0

x

Types of roots of a quadratic equation  Case

3: When b2 – 4ac < 0, the roots are complex. a>0

y

O The graph does not intersect the x-axis.

y

x

O

a<0

x

Relation between roots and coefficients  

 

Let  and  be the roots of the quadratic equation ax2 + bx + c = 0. Then, x = , or x= x -  = 0, or x-=0 (x - )(x - ) = 0 x2 – ( + )x +  = 0 This must be the same as the original eqation. Writing the original equation as x2 + (b/a)x + c/a = 0 and comparing coefficients, we have b c   ,   a a