solving quadratics

solving quadratics

Factorisation
1. Means putting it into 2 brackets
2. standard quadratic is ax² + bx + c = 0
3. EXAMPLE 1
  1. solve x² - x = 12
    1. x² - x - 12 = 0
      1. (x )(x ) = 0
        (x + 3)(x - 4)
        (x + 3) = 0 (x - 4) = 0
        x = -3 x = 4
        these are our answers
        we make each bracket equal zero to find the values
        We need to find two numbers that multiply to give c (which in this case is -12) and add to give b (which is 1)
        3 x -4 = -12 3 + -4 = -1
        This is a template for out brackets
      2. To solve, it must always equal zero
4. EXAMPLE 2
  1. solve 3x² + 7x -6 = 0
    1. in this case a (coefficient of x) is not 1
    2. tn = -18
      1. for this, we need two numbers that multiply to get -18 ( -6 x 3 ) , and add to get 7
      2. 9 and -2
        These are our two numbers, 9 x -2 = -18, and 9 + -2 = 7
        3x² + 9x -2x -6 = 0
        we replace the b value with our two numbers
        ,3x² + 9x, ,-2x -6,
        3x(x + 3)
        -2(x + 3)
        (3x -2) (x + 3)
        x = -3 x = 2/3 / 6.67
        you then reverse the brackets and thats your two answers
        The two on each outside and put them in a bracket, and the others in the bracket. these are out two factorised brackets
        we then use the same numbers in the brackets with the seconds two parts ( x+ 3), and then chose what these are multiplied by and put it on the outside
        we then find a common denominator between the first two parts, in this case, 3x. we then put the two numbers inside the brackets
        we then separate it into two parts, the first two bits, and the last two bits
Quadratic formula
1. -b ± √ b² - 4ac x=____________________ 2a
2. example:
  1. solve 3x² + 7x = 1
    1. 3x² + 7x -1 = 0
      1. x = -7 ±√ 7²-(4 x 3 x -1) _______________ 2 x 3
        = -7 ± √ 49 + 12 ___________ 6
        = -7+√61 _______ 6
        = -7-√61 _______ 6
        you now have two equations to put into the calculator
        x = 0.1350
        x = -2.468
        these are the two solutions
        4 x 3 x -1 = -12, 7² - -12 is 49 + 12
        then we sub in the values into the formula, a = 3, b = 7 and c = -1
      2. first put the equation into the standard format
completing the square
1. convert your equation into the standard format, ax² + bx + c = 0
  1. write out the initial bracket as (x + b/2)²
    1. multiply out the brackets and compare it to the original equation
      1. than add or subtract a number to make it identical to the original equation, add this value on after the (x + b/2)²
  2. -HOWEVER- if a ≠ 1, before all of these steps, once your equation is in the standard format divide it by a, and have a on the outside of the rest of the equation in a bracket to make the coefficient of x, 1. then continue with the rest of the steps
2. write 2x² + 5x + 9 in the form a( x + m)² + n
  1. 2x² + 5x + 9
    1. 2 (x² + 5/2 x) + 9
      1. 2 (x + 5/4)² = 2x² + 5x + 25/8
        9 = 72/8
        72/8 - 25/8 = 47/8
        2 (x + 5/4)² + 47/8
        putting that on the end of the initial bracket
        finding out the value to add/subtract to complete the square
        converting c to the same fraction ( to make it easier )
        this is the initial bracket, multiplied out
      2. take out a factor of two
    2. standard format
  2. Example:
Quadratic graphs
1. quadratic functions are the form y = anything with x²
  1. they all have a symmetrical bucket shape
2. to complete a table of values from an equation, you just sub in the x values, and get the y value to get a point on a graph
3. when sketching a graph, always label the important points on it
4. the solution to a quadratic are the x intercepts on a graph

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