solving quadratics

solving quadratics

1. Factorisation

   • Means putting it into 2 brackets
   • standard quadratic is ax² + bx + c = 0
   • EXAMPLE 1
     • solve x² - x = 12
       • x² - x - 12 = 0
         • (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
         • To solve, it must always equal zero
   • EXAMPLE 2
     • solve 3x² + 7x -6 = 0
       • in this case a (coefficient of x) is not 1
       • tn = -18
         • for this, we need two numbers that multiply to get -18 ( -6 x 3 ) , and add to get 7
         • 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

2. Quadratic formula

   • -b ± √ b² - 4ac x=____________________ 2a
   • example:
     • solve 3x² + 7x = 1
       • 3x² + 7x -1 = 0
         • 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
         • first put the equation into the standard format

3. completing the square

   • convert your equation into the standard format, ax² + bx + c = 0
     • write out the initial bracket as (x + b/2)²
       • multiply out the brackets and compare it to the original equation
         • than add or subtract a number to make it identical to the original equation, add this value on after the (x + b/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
   • write 2x² + 5x + 9 in the form a( x + m)² + n
     • 2x² + 5x + 9
       • 2 (x² + 5/2 x) + 9
         • 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
         • take out a factor of two
       • standard format
     • Example:

4. Quadratic graphs

   • quadratic functions are the form y = anything with x²
     • they all have a symmetrical bucket shape
   • 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
   • when sketching a graph, always label the important points on it
   • the solution to a quadratic are the x intercepts on a graph