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Taken from Haese Mathematics:Mathematics for the international

student: Mathematics SL

Chapter 1Quadratics

Alzbeta Bavorova

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Ch1 Quadratics = 0 The roots or solutions of = 0 are the values of x for which the equation is true.

FactorizationQuick if you have easy numbers. Tricky if a =/= 1 .

1. Rearrange equation to have zero on one side and the rest on the other.2. Divide by a .3. Find numbers s and r , for which:

a * c = s * r b = s + r

4. Rewrite equation as ( )( )= 0 5. x 1 = -s ; x 2 = -r

Completing the squarePerfect squares are expressions such as: (x + 1)2 , (x + 2) 2 => (x + a) 2

Use the formulas ( )= 2 and ( )= 2 4 1=0

1. Move c to the other side.

4 = 1

2. Pick a corresponding new c by dividing the b by 2 and squaring it.

= ()

= 4 3. Add c NEW to both sides

4 4 = 3 4. Factorize and solve

( 2) = 3

2=3

, = 3 2

Quadratic formulaFoolproof.

, = 4

2 , =

2

The discriminant () of a quadratic and sign diagrams = 4

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If:

> 0 .. Two real solutions

= 0 .. One solution

< 0 .. No real solution

A sign diagram (such as the one showed below) shows for what values is the quadratic positive, zero or negative.The points on the number line represent the roots of the equation.

Quadratic functions=

(If we substitute y by 0, we have a quadratic equation.)

The graph of a quadratic function is a parabola , which is one of the conic sections (produced by cutting the cone bya plane parallel to the cones slant side).

(http://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Parabola_features.svg/2000px-Parabola_features.svg.png )

If a > 0 , the parabola is concave/concave up (like the image).

If a < 0 , the parabola is convex/concave down .

http://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Parabola_features.svg/2000px-Parabola_features.svg.pnghttp://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Parabola_features.svg/2000px-Parabola_features.svg.pnghttp://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Parabola_features.svg/2000px-Parabola_features.svg.pnghttp://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Parabola_features.svg/2000px-Parabola_features.svg.png

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Quadratic form, a =/= 0 Factsy = a(x-p)(x-q) x-intercepts are p and q

axis of symmetry is

= + vertex is ( + , + )

y = a(x-h) 2 touches x-axis at h = 0 axis of symmetry is x = h vertex is (h, 0)

y = a(x-h) 2 + k axis of symmetry is x = h vertex is (h, k)

y = ax 2 + bx + c y-intercept is c axis of symmetry is

= vertex is ( , ) x-intercepts can be

calculated using 0 = ax 2

+bx + c

Sketching graphs by completing the square

Positive definite and negative definite quadraticsPositive definite quadratics are those which are positive for all values of x . This means they have no x-intercept anda > 0.

Negative definite quadratics are the exact opposite.

Finding a quadratic form its graphThe roots of the function are its x-intercepts.

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To get the value of the y-intercept, substitute x with 0.

The value of a tells us the orientation of the function.

The axis of symmetry will be the average of the x-intercepts.

Where functions meetFunction intercepts can be found by solving the functions simultaneously

: = : = = => =

Quadratic optimizationOptimization is the process of finding the minimum or maximum of a function. These equal to the y-value of the

vertex .