Download Quadratic Equations

Quadratic
Equations
MTED 301
May 7, 2008
Diane Yum
Quadratic Equation

An equation that could be written as
ax2+bx+c=0
Quadratic Equation

An equation that could be written as
ax2+bx+c=0
Standard form of a quadratic equation
ax2+bx+c=0
Quadratic Equation

An equation that could be written as
ax2+bx+c=0
Standard form of a quadratic equation
ax2+bx+c=0
- The quadratic is on the left and 0 is
on the right.
- Moreover, it is standard for the
leading coefficient A to be positive.
3 different ways to solve a
quadratic equation
3 different ways to solve a
quadratic equation
 Solve
by Factoring
3 different ways to solve a
quadratic equation
 Solve
by Factoring
 Solve by Completing the
Square
3 different ways to solve a
quadratic equation
 Solve
by Factoring
 Solve by Completing the
Square
 Solve by Using “Quadratic
Formula”
Solving by Factoring

To solve a quadratic equation,
Put all terms on one side of the equal sign,
leaving zero on the other side (Standard
Form)
② Factor
③ Set each factor equal to zero
④ Solve each of these equations
⑤ Check by inserting your answer in the
original equation
①
Example of factoring
Ex) Solve for y: y2 = -6y – 5
First, change the equation into the
standard form: y2 + 6y + 5 = 0
Factoring, (y+5) (y+1) = 0
Y+5 = 0 or y+1 = 0
Y = -5
or y = -1
Check your answer
(-5)2 = -6(-5) – 5 or (-1)2 = -6(-1) -5
25 = 30 – 5
1 = 6-5
25 = 25
1 = 1
You got it right 
Solving by Completing the Square


Completing the Square: Finding something
to add to a quadratic to make it a perfect
square
Expression: (x+k)2
Applying our formula for squaring a
binomial, we get
(x+k)2 = x2 + 2xk + k2
So if you have an expression of the form
x2+bx and you want to find something to
add to it to make it a perfect square,
then you need to
 Divide b by 2 to get k
 Square k to get k2
9 2 81
9
Ex) y2 – 9y
( ) 
92 
2
4
2
81
9 2
2
y  9y 
 (y  )
4
2
Example of Completing the Square
Ex) Solve x2 + 6x – 7 = 0 by completing the s
quare
1)
x2 + 6x – 7 = 0
2)
x2 + 6x
=7
3)
(6/2)2 = 9
4)
x2 + 6x + 9 = 7 + 9
5)
(x + 3 )2
= 16
6)
x+3
= +4, -4
7)
x
= -3 + 4 and -3 – 4
8)
x
= +1 and -7
Check your answer
x2
+6x –7=0
(1)2 + 6(1) – 7 = 0
1 + 6 –7=0
and
You got it right again 
x2 + 6x – 7 = 0
(-7)2 + 6(-7) – 7 = 0
49 – 42 - 7 = 0
Solving by Quadratic Formula
Quadratic Formula :
 b  b  4ac
x
2a
2
Easy Steps to solve by quadratic formula
1)
2)
3)
Find a, b, and c in the standard form
Substitute numbers of a, b, and c in the quadratic
formula
Find the value of x
Example of Quadratic Formula
Ex) Solve the equation of 2x2 + 5x = 10 by
using a quadratic formula
① Rewrite the equation into a standard form
2x2 + 5x – 10 = 0
② Identify the values of a, b, and c
a = 2, b = 5, c = -10
③Substitute these values into the Quadratic Formula
Substitution
 5  (5)  4(2)(10)  5  25  80  5  105
x


2(2)
4
4
2
 5  105
 5  105
x
and
4
4
You can substitute the x values into the o
riginal equation to check the answer!
Homework
Due : Next Class Meeting
Solve each equation by factoring,
completing the square, or the
quadratic formula.





Solve (x+1)(x-3) = 0
2
Solve x + x – 4 = 0
Solve x2 – 3x – 4 = 0
2
Solve 6x + 11x – 35 = 0
Solve x2 – 48 = 0