Solving Quadratics

Sometimes solving quadratics is easy

Sometimes you recognize a form

Sometimes you can factor

But no matter what,You can ALWAYS use the Quadratic Formula

Example

What does the QF say?

What does the QF say?

What does the QF say?

QF says

QF says

Location of x-intercept,

“roots” or “zeros” of the parabola

Line of symmetry,Location of the vertex,

Location of the max or min

Synonyms

SynonymsLine of symmetryLocation of vertexLocation of extremum

(max or min)

VertexExtremum

x-interceptrootzero

x-interceptrootzero

Why does the QF work?

ax2

ax

x bxx

b

+ = D

Stretch everything by a

(ax)2

ax

ax abxax

b

+ =

aD

Split b in half

(ax)2

ax

ax abx/2ax

b/2

+ =

aDabx/2

b/2

Rearrange

(ax)2

ax

ax abx/2

b/2

=

aD

abx/2b/2

Complete the square

(ax)2

ax

ax abx/2

b/2

=

aD

abx/2b/2

b2

/4

b2

/4+

Reorganize

(ax+b/2)2

ax+b/2

ax+b/2

=

aDb2

/4+

Equationify

(ax+b/2)2

ax+b/2

ax+b/2

=

aDb2

/4+

Rearrange

(ax+b/2)2

ax+b/2

ax+b/2

=

aDb2

/4+

Square root

(ax+b/2)2

ax+b/2

ax+b/2

=

aDb2

/4+

Rearrange

(ax+b/2)2

ax+b/2

ax+b/2

=

aDb2

/4+

-b/2 from both sides

(ax+b/2)2

ax+b/2

ax+b/2

=

aDb2

/4+

/a on both sides

(ax+b/2)2

ax+b/2

ax+b/2

=

aDb2

/4+

But what happened to c?

But what happened to c?

ax2

ax

x bxx

b

+ = D

But what happened to c?

ax2

ax

x bxx

b

+ - D =0

But what happened to c?

ax2

ax

x bxx

b

+ - D =0

+c=-d

The Quadratic Formula

ax2 bx+ + =0c

Solve: x2+9x+8=0. Select the most correct answer below!

A) x=1, x=8B) x= -1, x= -8C) x= -1, x = 8D) x=1, x= -8 E) No real solutions

Solve: x2+9x+8=0. Select the most correct answer below!

B

Find the zeros of f(x)=x2+4x+2

A) -2 ± 2sqrt(2)B) -2 ± sqrt(2)C) -2 ± sqrt(8)D) 2 ± 2sqrt(2) E) No real solutions

Find the zeros of f(x)=x2+4x+2

B

Counting Roots

(x-2)(x-4) has two real roots:x=2 and x=4.

Counting Roots

(x-3)(x-3) has two real roots:x=3 and x=3.Both roots are in the same place,But it is useful to think of them astwo roots.

Counting Roots

(x-(3-i))(x-(3+i)) has two complex roots:x=3-i and x=3+i.

Counting roots

• A quadratic always has exactly two roots– Sometimes the roots are the same– Sometimes the roots are complex

• A quadratic always has an even number of complex roots.– Possible roots are: two real, or two complex. You

can never have 1 real and 1 complex

Why?

A quadratic turns and continues infinitely.

Because of this, if the quadratic crosses the x axis once, it HAS to cross a second time.

Always zero or two real roots.

Consider the quadratic function f(x)=x2+2x+5.Which of the following statements is true?

A) f(x) has 1 real zero and 1 complex zero.

B) f(x) has no real zeros.C) f(x) has 2 real zeros.D) f(x) has 3 real zeros.E) None of the above are true.

Consider the quadratic function f(x)=x2+2x+5.Which of the following statements is true?

B) No real zeros