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Angstrom Care 培苗社 Quadratic Equation II

Quadratic Equations, Quadratic Functions and Absolute Values

Solving a Quadratic Equation
by factorization by graphical method by taking square roots by quadratic equation by using completing square

By factorization roots (solutions)

By graphical method y roots x O

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

By taking square roots

Solving a Quadratic Equation by the quadratic Formula

By quadratic equation

a = 1 b = -7 c = 10

In general, a quadratic equation may have :
(1) two distinct (unequal) real roots (2) one double (repeated) real root (3) no real roots

Two distinct (unequal) real roots
x-intercepts

One double (repeated) real roots
x-intercept

No real roots no x-intercept

Nature of Roots

△ = 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.

Two distinct (unequal) real roots
△ = b2 - 4ac > 0 x-intercepts

One double (repeated) real roots
△ = b2 - 4ac = 0 x-intercept

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

Solving a Quadratic Equation by Completing the Square

Solving a Quadratic Equation by Completing the Square

Relations between the Roots and the Coefficients

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

Forming Quadratic Equations with Given Roots

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

Linear Functions and Their Graphs

y c＞0 x O

y x O c＜0

Linear Functions

y m＞0 c＞0 c x O

y m＞0 c＜0 x O c

y c x O m＜0 c＞0

y x O c m＜0 c＜0

y O x c m＜0 c＝0

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

Vertex Open downwards Line of symmetry (a＞0)

Vertex (Turning point)
Local (Relative) Maximum point (max. pt.) Local (Relative) Minimum point point (mini. pt.)

y = ax2

y y = ax2 (a＞0) x O

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

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

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

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)

Absolute Values

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

For all real numbers x and y,

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

Thank you

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