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**Angstrom Care 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 and Absolute Values**

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**Solving a Quadratic Equation**

by factorization by graphical method by taking square roots by quadratic equation by using completing square

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By factorization roots (solutions)

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By graphical method y roots x O

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**? By taking square roots A quadratic equation must contain two roots.**

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By taking square roots

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**Solving a Quadratic Equation by the quadratic Formula**

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By quadratic equation

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a = 1 b = -7 c = 10

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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 - 4ac Since 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**

△ = b2 - 4ac > 0 x-intercepts

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**One double (repeated) real roots**

△ = b2 - 4ac = 0 x-intercept

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No real roots △ = b2 - 4ac < 0 no x-intercept

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**Solving a Quadratic Equation by Completing the Square**

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**Solving a Quadratic Equation by Completing the Square**

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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 and product of roots = αβ

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**Forming Quadratic Equations with Given Roots**

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**x = 2 or x = -3 x – 2 = 0 or x + 3 = 0 (x – 2)(x + 3) = 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 x2 – (sum of the roots)x + (product of roots) = 0

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**Linear Functions and Their Graphs**

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y c＞0 x O

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y x O c＜0

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Linear Functions

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y m＞0 c＞0 c x O

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y m＞0 c＜0 x O c

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y c x O m＜0 c＞0

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y x O c m＜0 c＜0

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y O x c m＜0 c＝0

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**Open upwards Open upwards (a＞0) Vertex Line of symmetry**

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Vertex Open downwards 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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y y = ax2 (a＞0) x O

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**y y = ax2 + bx + c b2 - 4ac＞0 2 real roots (a＞0) (c＜0) x roots O**

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**y y = ax2 + bx + c b2 - 4ac＝0 repeated roots (a＞0) (c＞0) x root O**

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**y y = ax2 + bx + c B2 - 4ac＜0 No real roots (a＞0) (c＞0) x No intercept**

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**mini value of the function = -1 mini point = (-2, -1)**

Finding the turning point of a Quadratic Function by Completing the Square Because a = +ve, there exists a minimum point. mini value of the function = -1 mini point = (-2, -1)

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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 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,**

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**If | x | = a, where a ≧0, then x = a or x = -a**

Generalization If | x | = a, where a ≧0, then x = a or x = -a

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Thank you

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