solving quadratics
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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