Download Objective: Students will be able to write a quadratic equation with

Objective: Students will be able to write a quadratic equation with given roots
and solve a quadratic equation by factoring.
The intercept form of a quadratic equation is:
y=a(x–p)(x–q)
In this equation, p and q represent the x-intercepts.
To change an equation in intercept form back into standard form, you just need
to use FOIL!
Example 1: Write a quadratic equation with 4 and -2 as its roots. Write the
equation in standard form.
Example 2: Write a quadratic equation with ½ and -6 as its roots. Write the
equation in standard form with a, b, and c as integers.
When we are given an equation in standard form, we can factor the quadratic
into the product of two binomials.
We have seen factoring before…
The sum of these numbers is b
x2+bx+c = (x +
)(x +
)
The product of these numbers is c
Example 3: Factor x2+6x+5
If we are to solve a quadratic by factoring, we have to keep in mind that the
solutions are when y=0!
So we look at the zero property….
This says if a*b=0, then a=0, b=0 or they are both 0!
Example 4: Find the solutions to x2+6x+5=0 by factoring.
For each quadratic, find the solutions by factoring.
1. x2+12x+32 = 0
11. x2 + 20x + 99=0
2. x2+13x+42 = 0
12. x2 - 3x – 88=0
3. x2-5x+6 = 0
13. x2 + 11x + 30=0
4. x2-4x-60 = 0
14. x2 + x – 30=0
5. x2-7x-12=0
15. x2 + 8x – 9=0
6. x2-9x-10=0
16. x2 + 6x – 72=0
7. x2+5x-50=0
17. x2 + 10x + 24=0
8. x2 - 14x + 45=0
18. x2 - 5x + 4=0
9. x2 - 18x + 80=0
19. x2 + 8x – 20=0
10. x2 - 6x + 9=0
20. x2 + 2x – 35=0
What pattern do you notice with the signs?
If x2+bx+c=0 …..then (x
)(x
)
If x2 –bx+c=0 …….then (x
)(x
)
If x2+bx–c=0 …….then (x
)(x
)
If x2 –bx–c=0 …….then (x
)(x
)
Homework 5-3 Factoring a Quadratic Equation
1. Write a quadratic equation with -2 and -6 as its roots. Write the equation
in standard form with a, b, and c as integers.
2. Write a quadratic equation with 5 and 5 as its roots. Write the equation in
standard form with a, b, and c as integers.
3. Write a quadratic equation with -10 and 10 as its roots. Write the
equation in standard form with a, b, and c as integers.
4. Factor each quadratic equation to find the solutions.
a. x2 + 17x + 60 = 0
f. x2 - 14x + 33 = 0
b. x2 - 10x + 16 = 0
g. x2 - 3x – 70 = 0
c. x2 - 7x + 6 = 0
h. x2 - 16x + 55 = 0
d. x2 + 3x – 18 = 0
i. x2 + 5x – 50 = 0
e. x2 - 16x + 48 = 0
j. x2 + 6x – 16 = 0
Start-Up - Quadratic Functions
1. Graph the function f(x)=x2+10x+16
a. What is the equation for the line of symmetry?
b. What is the vertex?
c. What are the solutions?
2. Write a quadratic equation with 8 and -4 as the solutions.
3. Find the solutions by factoring to the function f(x)=x2-7x+12
4. What are the roots of the graph below?
What happens if a  1 or  1 ?
1. How can we factor 2 x 2  6 x  4  0 ?
We can factor this quadratic by first finding the GCF (___________________)!
In this example the GCF=________.
2. How can we factor x 2  6 x  0 ?
In this example the GCF=_________.
3. How can we factor 5 x 2  15 x  0
In this example the GCF=_________.
Factor each equation to find the solutions.
1. 3x2 - 3x = 0
6. 4x2-4x = 0
2. 33x2 + 363x = 0
7. 10x2-20x+10 = 0
3. -33x2 - 330x = 0
8. x2-14x = 0
4. 12x2 – 5x = 0
9. x3+7x2+12x = 0
5. 3x3+18x2+24x = 0
10. 4x2 – 48x = 0
Mixed practice: Solve for x in each equation.
11. x2+100=-20x
13. 5x2+3x+2=0
12. 2x2 – 16x – 18 = 0
14. x2 – 8x=20
20. x2 = 11x
15. 6x2 + 66x + 180=0
21. (x-5)(x+6)=0
16. x2-7x=0
22. x2 + 48 = 16x
17. -3x2 – 15x + 150=0
23. -11x2 + 66x = 88
18. x2 + x = 90
24. x2 - 7x + 6 = 0
19. 4x2-36x=0
25. x2 – 100 = 0
When I have a quadratic expression, the first thing I look to do is
____________________________!
HOMEWORK 5-3 FACTORING A GCF
1. 2x2 + 28x + 98 =0
2. x2 + 20 =12x
3. -x2 + 19x – 84=0
4. 4x2 -4 x – 48=0
5. x2+10x=0
6. 3x2+9x=0
7. 2x2+12x=80
8. Write a quadratic equation in standard form with 6 and -8 as the roots.
9. Write a quadratic equation in standard form with -6 and 8 as the roots.
10. What is the vertex of the equation f(x) = x2-18x+72
Start-Up
Match each equation to its factors
1.
2.
3.
4.
5.
x2+7x+10
2x2+4x
4x2+24x+36
x2+14x-120
x2-8x+15
A.
B.
C.
D.
E.
4(x+3)2
(x+20)(x – 6)
(x – 5)(x – 3)
2x(x+2)
(x+2)(x+5)
Start-Up
Match each equation to its factors
1.
2.
3.
4.
5.
x2+7x+10
2x2+4x
4x2+24x+36
x2+14x-120
x2-8x+15
A.
B.
C.
D.
E.
4(x+3)2
(x+20)(x – 6)
(x – 5)(x – 3)
2x(x+2)
(x+2)(x+5)
Name
General Form
Difference of
Perfect
Squares
Example
x2 – 25 =0
Factored Form
a 2 - b2 =
(a+b)(a – b)
x3 + 27 = 0
Sum of
Perfect Cubes
a3 + b 3 =
(a + b)(a2 – ab + b2)
x3 – 8 = 0
Difference of
Perfect Cubes
a3 – b 3 =
(a – b)(a2 + ab + b2)
x2+10x+25=0
Perfect
Square
Trinomial
a2+2ab+b2 =
(a+b)(a+b)
Examples: Factor each expression and solve for x.
1. x2 – 9 = 0
4. x2 - 20x + 100 = 0
2. x2 - 18x + 81 = 0
5. x2 – 81 = 0
3. x3 – 1000 = 0
6. x2 - 14x + 49 = 0
10. x2 – 49 = 0
7. x3 + 125 = 0
11. x3-64=0
8. x2 – 100 = 0
12. x2 + 18x + 81 = 0
9. x2 + 22x + 121 = 0
13. x2 – 36 = 0
Some harder cases…Factor each expression and solve for x.
1. 100x2 – 64 = 0
4. 125x3 – 64 = 0
2. 4x2 – 16 = 0
5. 16x2 – 48x+36 = 0
3. 8x3 + 27 = 0
HOMEWORK – 5-3 Factoring Special Cases
Factor and solve for x.
1. 27x3 + 1=0
6. x2 + 4x + 4=0
2. x2 + 10x + 25=0
7. 9x2 + 24x + 16=0
3. x3 + 1=0
8. 49x2 + 28x + 4=0
4. x3 – 1=0
9. x2-64=0
5. x3 + 8 =0
10. x2-144=0
11. While standing on the roof outside room 227, you throw a ball in the air.
The equation h(t)=-11t2 +44t+55 describes the height, h, of the ball after
t seconds.
a) What is “c”? What does it represent?
b) What is the maximum height reached by the ball?
c) At what time will the ball be exactly even with you?
d) At what time will the ball land?