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Angstrom CareAngstrom Care培苗社培苗社
Quadratic Equation II
積極推崇『我要學』的心態,
糾正『要我學』的被動心態。
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Quadratic Equations,Quadratic Functions andAbsolute Values
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Solving a Quadratic Equationby factorization by graphical methodby taking square rootsby quadratic equation by using completing square
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By factorization
01072 xx0)2)(5( xx
02__05 xorx2__5 xorx
roots (solutions)
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By graphical method
01072 xx
x
y
O
roots
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By taking square roots
4)32( 2 x432 x
232 x52 x
5.2xA quadratic equation must contain two roots.
?
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By taking square roots
4)32( 2 x
432 x
232 x
152 orx 5.05.2 orx
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Solving a Quadratic Equation by the quadratic Formula
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By quadratic equation
0,02 acbxaxIf
a
acbbx
2
42
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)1(2
)10)(1(4)7()7( 2 x
01072 xx
a = b = c =1 10-7
25 xorx
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In general, a quadratic equation may have :
(1) two distinct (unequal) real roots
(2) one double (repeated) real root
(3) no real roots
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Two distinct (unequal) real roots
x-intercepts
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One double (repeated) real roots
x-intercept
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No real roots
no x-intercept
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Nature of Roots
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△ = b2 - 4acSince the expression b2 - 4ac can be used to determine the nature of the roots of a quadratic equation in the form ax2 – bx + c = 0, it is called the discriminant of the quadratic equation.
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Two distinct (unequal) real roots
x-intercepts
△ = b2 - 4ac > 0
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One double (repeated) real roots
x-intercept
△ = b2 - 4ac = 0
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No real roots
no x-intercept
△ = b2 - 4ac < 0
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Solving a Quadratic Equation by Completing the Square
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Solving a Quadratic Equation by Completing the Square
0342 xx
03)2
4()
2
4(4 222 xx
034)2( 2 x
1)2( 2 x
12 x
2121 orx31 orx
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Relations between the Roots and the Coefficients
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If α and β(p and q, x1 and x2) are the roots of ax2 + bx + c = 0,then sum of roots = α + β
a
b
and product of roots = αβ
a
c
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Forming Quadratic Equations with Given Roots
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x2 – (sum of the roots)x + (product of roots) = 0
Forming Quadratic Equations with Given Roots
In S.3, when α = 2 and β = -3
x = 2 or x = -3
x – 2 = 0 or x + 3 = 0
(x – 2)(x + 3) = 0
x2 + x – 6 = 0
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Linear Functions and Their Graphs
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c > 0
x
y
O
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c < 0
x
y
O
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c >0
x
y
O
m > 0
c
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c <0 x
y
O
m > 0
c
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c >0
x
y
O
m < 0
c
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c <0
x
y
O
m < 0c
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c =0
x
y
O
m < 0c
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Open upwards
Vertex
Open upwards
Line of symmetry
(a >0)
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Open downwards
Vertex
Line of symmetry (a >0)
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Vertex (Turning point)
Local (Relative) Maximum point (max. pt.)
Local (Relative) Minimum point point (mini. pt.)
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y = ax2
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x
y
O
y = ax2
(a > 0)
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x
y
O
y = ax2 + bx + c
(a > 0)b2 - 4ac > 0
roots
2 real roots(c < 0)
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x
y
O
(a > 0)
b2 - 4ac = 0
repeated roots
root
y = ax2 + bx + c
(c > 0)
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x
y
O
(a > 0)
B2 - 4ac < 0
No real roots
y = ax2 + bx + c
(c > 0)
No intercept
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Finding the turning point of a Quadratic Function by Completing the Square
0342 xx
03)2
4()
2
4(4 222 xx
034)2( 2 x
01)2( 2 x
mini value of the function = -1mini point = (-2, -1)
Because a = +ve, there exists a minimum point.
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Absolute Values
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Let x be any real number. The absolute value of x, denoted by | x |, is defined as x
x if x 0.≧
-x if x < 0.
eg. | 5 | = 5, | 0 | = 0, | -5 | = 5
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For all real numbers x and y,
xx 22
xx yxyx
y
x
y
x (y ≠ 0)
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Generalization
If | x | = a, where a ≧0, then x = a or x = -a
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Thank you
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