3.3      Solve  :    11x+2 = 0  Subtract  2  from both sides of the equation :                       11x = -2 Divide both sides of the equation by 11:                     x = -2/11 = -0.182

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2.1     Factoring  11x2-42x-8  The first term is,  11x2  its coefficient is  11 .The middle term is,  -42x  its coefficient is  -42 .The last term, "the constant", is  -8 Step-1 : Multiply the coefficient of the first term by the constant   11 • -8 = -88 Step-2 : Find two factors of  -88  whose sum equals the coefficient of the middle term, which is   -42 .

3/4idx 1 3/4odbearing

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :                      11*x^2-42*x-(8)=0

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4.1      Find the Vertex of   y = 11x2-42x-8Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 11 , is positive (greater than zero).  Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.  Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.  For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   1.9091   Plugging into the parabola formula   1.9091  for  x  we can calculate the  y -coordinate :   y = 11.0 * 1.91 * 1.91 - 42.0 * 1.91 - 8.0 or   y = -48.091

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -44  and  2                      11x2 - 44x + 2x - 8Step-4 : Add up the first 2 terms, pulling out like factors :                    11x • (x-4)              Add up the last 2 terms, pulling out common factors :                    2 • (x-4) Step-5 : Add up the four terms of step 4 :                    (11x+2)  •  (x-4)             Which is the desired factorization

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Root plot for :  y = 11x2-42x-8 Axis of Symmetry (dashed)  {x}={ 1.91}  Vertex at  {x,y} = { 1.91,-48.09}   x -Intercepts (Roots) : Root 1 at  {x,y} = {-0.18, 0.00}  Root 2 at  {x,y} = { 4.00, 0.00}

3.1    A product of several terms equals zero.  When a product of two or more terms equals zero, then at least one of the terms must be zero.  We shall now solve each term = 0 separately  In other words, we are going to solve as many equations as there are terms in the product  Any solution of term = 0 solves product = 0 as well.

4.2     Solving   11x2-42x-8 = 0 by Completing The Square . Divide both sides of the equation by  11  to have 1 as the coefficient of the first term :   x2-(42/11)x-(8/11) = 0Add  8/11  to both side of the equation :    x2-(42/11)x = 8/11Now the clever bit: Take the coefficient of  x , which is  42/11 , divide by two, giving  21/11 , and finally square it giving  441/121 Add  441/121  to both sides of the equation :  On the right hand side we have :   8/11  +  441/121   The common denominator of the two fractions is  121   Adding  (88/121)+(441/121)  gives  529/121   So adding to both sides we finally get :   x2-(42/11)x+(441/121) = 529/121Adding  441/121  has completed the left hand side into a perfect square :   x2-(42/11)x+(441/121)  =   (x-(21/11)) • (x-(21/11))  =  (x-(21/11))2 Things which are equal to the same thing are also equal to one another. Since   x2-(42/11)x+(441/121) = 529/121 and   x2-(42/11)x+(441/121) = (x-(21/11))2 then, according to the law of transitivity,   (x-(21/11))2 = 529/121We'll refer to this Equation as  Eq. #4.2.1  The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of   (x-(21/11))2   is   (x-(21/11))2/2 =  (x-(21/11))1 =   x-(21/11)Now, applying the Square Root Principle to  Eq. #4.2.1  we get:   x-(21/11) = √ 529/121 Add  21/11  to both sides to obtain:   x = 21/11 + √ 529/121 Since a square root has two values, one positive and the other negative   x2 - (42/11)x - (8/11) = 0   has two solutions:  x = 21/11 + √ 529/121    or  x = 21/11 - √ 529/121 Note that  √ 529/121 can be written as  √ 529  / √ 121   which is 23 / 11

4.3     Solving    11x2-42x-8 = 0 by the Quadratic Formula . According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :                                                 - B  ±  √ B2-4AC  x =   ————————                      2A   In our case,  A   =     11                      B   =   -42                      C   =   -8 Accordingly,  B2  -  4AC   =                     1764 - (-352) =                      2116Applying the quadratic formula :                42 ± √ 2116    x  =    ——————                      22Can  √ 2116 be simplified ?Yes!   The prime factorization of  2116   is   2•2•23•23  To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).√ 2116   =  √ 2•2•23•23   =2•23•√ 1   =                ±  46 • √ 1   =                ±  46 So now we are looking at:           x  =  ( 42 ± 46) / 22Two real solutions:x =(42+√2116)/22=(21+23)/11= 4.000 or:x =(42-√2116)/22=(21-23)/11= -0.182