Transcript The Quadratic Formula

Solutions to quadratic equations
2x2+7x+6
(2A.8) Quadratic and square root
functions. The student formulates
equations and inequalities based on
quadratic functions, uses a variety of
methods to solve them, and analyzes the
solutions in terms of the situation.
The student is expected to:
(B) analyze and interpret the solutions of
quadratic equations using discriminants and
solve quadratic equations using the quadratic
formula;
The Babylonians
Stopping a car
Every polynomial has as many roots as its degree
A quadratic equation is a second degree polynomial,
it has two roots
x2 − 4
20
15
10
5
-4
-2
2
4
x2 − 4
20
15
10
5
-4
-2
2
(x+2)(x−2)
4
x2 +2x +1
25
20
15
10
5
-4
-2
2
4
x2 +2x +1
25
20
15
10
5
-4
-2
2
(x+1)2
4
x2 +2x +5
25
20
15
10
5
-4
-2
2
4
x2 +2x +5
25
20
15
10
5
-4
-2
2
4
(x+1+2i)(x+1−2i)
ax2+bx+c=0
The quadratic formula works for any quadratic
equation because it has already done the
steps of completing the square for you. All
you have to do to use it is substitute the
coefficients into the formula and simplify.
x = −b±√(b2 − 4ac)
2a
1. Get the equation in ax2+bx+c=0 form
2. Label the coefficients a,b and c then
substitute into the formula and simplifiy.
x = −b±√(b2 − 4ac)
2a
4x2+x=3
100
80
60
40
20
-4
-2
2
4
4x2+x−3=0
a=4, b=1, c=−3
4x2+x−3=0
a=4, b=1, c=−3
x=−1±√12 − 4(4)(−3)
2(4)
4x2+x−3=0
a=4, b=1, c=−3
x=−1±√1 + 48
8
4x2+x−3=0
a=4, b=1, c=−3
x=−1±√49
8
4x2+x−3=0
a=4, b=1, c=−3
x=−1±7
8
4x2+x−3=0
a=4, b=1, c=−3
x=−1+7
x=−1−7
8
8
4x2+x−3=0
a=4, b=1, c=−3
x=6
x=−8
8
8
4x2+x−3=0
a=4, b=1, c=−3
x=3
x=−1
4
9x2 −30x+25=0
250
200
150
100
50
-4
-2
2
4
9x2 −30x+25=0
a=9, b=−30, c=25
9x2 −30x+25=0
a=9, b=−30, c=25
x=30±√302 − 4(9)(25)
2(9)
9x2 −30x+25=0
a=9, b=−30, c=25
x=30±√900 − 900
18
9x2 −30x+25=0
a=9, b=−30, c=25
x=30±0
18
9x2 −30x+25=0
a=9, b=−30, c=25
x=30
18
9x2 −30x+25=0
a=9, b=−30, c=25
x=5
3
Exercises – solve using the quadratic formula
1. x2+5x+6=0
2. x2−6x=−8
3. 2x2+3x=0
4. 2x2−7x−4=0
5. x2+x=12
6. 3x2+x−2=0
7. x2−9=0
8. x2+2x−8=0
To memorize the quadratic formula
for musical intelligences
put these words to Pop Goes the Weasel
Minus bee plu-us or minus the square root of bee
squared
minus four a-a-ay cee
all over two ay.
For bodily-kinesthetic learners
write the quadratic formula before each
homework problem
Before a test
read the quadratic formula
before going into class
write it down when you first
get the test
A Visual Quadratic Equation
x2 +
−
x =
±
x+
=0
√
−
The discriminant tells you how many and what
type of solutions a quadratic equation has.
A quadratic equation is a second degree equation
so it can have at most two solutions
The discriminant is
b2 − 4ac
How many solutions did you get, what type of
solutions were they and what was the sign of
the discriminant in the equation
4x2+x=3
How many solutions did you get, what type of
solutions were they and what was the sign of
the discriminant in the equation
9x2 −30x+25=0
How many solutions did you get, what type of
solutions were they and what was the sign of
the discriminant in the equation
x2 −3x=−10
Equation
Type
4x2+x=3
real
Discriminant
#Solutions
+
2
9x2 −30x+25=0
real
0
1
x2 −3x=−10
complex
−
2
If the discriminant is positive
you get two real solutions
If the discriminant is 0
you get one real solution
If the discriminant is negative
you get two complex solutions
Exercises – describe the solutions using the
discriminant
1. 4x2−20x+25=0
2. 5x2−6x=-2
3. 10x2=x+2
4. x2+x−1=0