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Notes for IB maths SL - chapter 1, quadratics
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   Taken from Haese Mathematics: Mathematics for the international student: Mathematics SL Chapter 1 Quadratics Alzbeta Bavorova  Ch1 – Quadratics   =0   The roots  or solutions of   =0  are the values of x for which the equation is true. Factorization Quick 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 square Perfect 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 formula Foolproof.  , =±√  42  , = ±√ ∆2   The discriminant (Δ) of a quadratic  and sign diagrams ∆ =   4    If: Δ > 0 ……………….. Two real solutions   Δ = 0 ……………….. One solution   Δ < 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 by a plane parallel to the cone’s 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 .  Quadratic form, a =/= 0 Facts y = 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    Δ = 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 quadratics Positive definite   quadratics  are those which are positive for all values of  x  . This means they have no x-intercept and a > 0. Negative definite quadratics  are the exact opposite. Finding a quadratic form its graph The roots  of the function are its x-intercepts.
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