Download Section 1-6 - MrFridgesMathClassroom

Warm-Up
1
 45
2
1
 3i  2
3
Solve for x : x 2  3x  4  0
4
How can you determine whether two lines are parallel
or perpendicular or neither?
Section 1-6
Solving Quadratic Equations
Quadratic Equations
Definition: Any equation that can be written in the form
y = x2 − x − 2
ax 2  bx  c  0
Where a ≠0 (For if a = 0, the equation becomes a linear equation.) , is called a
quadratic equation. (Note: The letters a, b, and c are called coefficients)
Note: the actual function is written as
y  ax 2  bx  c
Definition: A root or solution, of a quadratic equation is a value of the variable that
satisfies the equation (i.e. what x value gives me zero?)
Want to solve for x :
ax 2  bx  c  0
3 ways to solve: (1) Factoring (2) Completing the square (3) the quadratic formula
Factoring

Recall: want to solve

ax 2  bx  c  0
IDEA - rewrite this equation as the product of
two factors (
)*(
)=0 (this is called factoring)


Now if at least one factor is zero, the equation is true
Note: the quadratic equation must be written in
standard form ax 2  bx  c  0 before it can be solved by
factoring.
Ex) Solve for x : x 2  5x  8  2
Factoring Tips
When trying to factor
ax 2  bx  c  0
, that is
in the form (
)*(
)=0, multiply a*c and write out the factors – these
factors must add to give b.
Ex) Solve for x : x 2  5 x  6  0
6*1=6
(-6)(-1)=6
2*3=6
(-2)(-3)=6
Ex) Solve for x : (3x - 2)(x  4)  11
More Examples of Factoring
Solve 3x2 -4x -7 =0
(
)(
) = 0 Find factors of 3 and -7
X2 -5x +4 = 0
(
)(
) =0 find factors of 4
6x2 – x - 2 =0
(
)(
) Find factors of 6 and -2
x2 + 8x + 15 =0
(
)(
) find factors of 15
Completing the Square

Recall: want to solve

ax 2  bx  c  0
IDEA: make one side a perfect square (that is
(x+w)2=___. We can then take the square root of both
sides and solve for x.

Note: before completing the square we must make the coefficient
of x2 1.
Ex) Solve for x : 2 x 2  12 x  7  0
Try to factor:
2*(-7)=-14
(-2)*7=-14
-14*1=-14
(-1)*14=-14
More Examples of Completing
the Square
x  3x  2  0
2
2
3
3
x  3 x     2   
2
2
2
3 2
(x  ) 
2
8 9
- 
4 4
2
9
9
x  3 x    = -2 +  
4
4
2
3
3
9
( x  )( x  )  2 
2
2
4
3
1
(x  ) =
2
4
1
6
x =

4
4
7
x =
4
More Examples of Completing
the Square
Let’s Solve a few examples
x
 4x 1  0
2
2y
2
7y 9  0
y
2
 7 y  12  0
y
2
 6y 3  0
3x
2
 5 x  10  0
Quadratic Formula
The roots of the equation ax 2  bx  c  0 are given by :
 b  b 2  4ac
x
2a
(a  0)
Ex) Solve for x : 2 x 2  7  4 x
Quadratic Equations
More Examples
Let’s Solve a few examples
1. x  6 x  4  0
2
2.x  2 x  15  0
2
3. y  7 y  30
2
4.
2t  3t  2  0
5.
5m 2  3m  2  0
6.
3p  8 p  5
2
2
Quadratic Formula – the Discriminant
The roots of the equation ax 2  bx  c  0 are given by :
 b  b 2  4ac
x
2a
(a  0)
The discrimina nt (b 2  4ac) determines whether t he solution w ill
be real or imaginary.
Because of this “discriminating ability,” b2 - 4ac is
called the discriminant.
If b2-4ac < 0, there are two conjugate imaginary roots
If b2-4ac = 0, there is one real root (called a double root)
If b2-4ac < 0, there are two different real roots
Selecting a Method to Solve a Quadratic
Equation
ax 2  bx  c  0
Situation
Method to use
a, b, and c are integers and b2 - 4ac is
a perfect square
i.e. b 2  4ac is an integer
Factoring
The equation has the form:
x2 + (even number)x + constant = 0
Completing the Square
All other cases
Quadratic Formula
Losing and Gaining Roots

We must be careful when solving
quadratic equations because we can


Lose roots (we don’t find all the roots)
and we can gain roots
Gain roots (we find roots that don’t
actually exist)
Losing Roots

Roots (solutions) can be lost when dividing both sides of an
equation by a common factor

Ex) 4 x( x  1)  3( x  1) 2
4 x ( x  1) 3( x  1) 2

(divide both sides by (x - 1)
( x  1)
( x  1)
Notice: x=1 is a solution, for when 1
4 x  3( x  1)
4 x  3x  3
x  3
is input for x, we get 0=0
• so we lost the root x=1 when we
divided by (x-1)
How to prevent this:

1.
2.
Bring all terms to one side and solve for zero
Solve by cases
Losing Roots
4 x( x  1)  3( x  1)
2

Bring all terms to one side and solve for zero

Solve by cases:

If x-1=0 (x=1)

If x-1≠0 (x≠1)
Gaining Roots

We can gain roots that don’t actually exist by multiplying both
sides of an equation by an expression or by squaring both sides
of an equation

Multiplying both sides of an equation by an expression


Ex)
Solve :
x  2 x  2 8  4x

 2
x2 x2 x 4
Squaring both sides of an equation

Ex)
Solve :
x6  x
Homework


p35: 1, 3, 5, 7, 11, 13, 16, 17,18, 20,
25, 27,30
Extra Credit: 35 and 44