Find the roots of the following quadratic equation by splitting the middle term x2+20x+99=0​

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Find the roots of the following quadratic equation by splitting the middle term x2+20x+99=0​

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Charlie 4 weeks 2021-08-16T13:20:48+00:00 1 Answer 0 views 0

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    2021-08-16T13:22:28+00:00

    Reformatting the input :

    Changes made to your input should not affect the solution:  

    “x2”   was replaced by   “x^2”.

    Step by step solution :-

    STEP  1  :  Trying to factor by splitting the middle term

    Factoring  x2+20x+99  

     

    The first term is,  x2  its coefficient is  1 .  

    The middle term is,  +20x  its coefficient is  20 .  

    The last term, “the constant”, is  +99  

     

    Step-1 : Multiply the coefficient of the first term by the constant   1 • 99 = 99  

    Step-2 : Find two factors of  99  whose sum equals the coefficient of the middle term, which is   20 .  

     

         -99    +    -1    =    -100  

         -33    +    -3    =    -36  

         -11    +    -9    =    -20  

         -9    +    -11    =    -20  

         -3    +    -33    =    -36  

         -1    +    -99    =    -100  

         1    +    99    =    100  

         3    +    33    =    36  

         9    +    11    =           20    That’s it

    Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  9  and  11  

                        x2 + 9x + 11x + 99  

     

    Step-4 : Add up the first 2 terms, pulling out like factors :  

                       x • (x+9)  

                 Add up the last 2 terms, pulling out common factors :  

                       11 • (x+9)  

    Step-5 : Add up the four terms of step 4 :  

                       (x+11)  •  (x+9)  

                Which is the desired factorization

    Equation at the end of step  1  :  

     (x + 11) • (x + 9)  = 0

    STEP  2  :  Theory – Roots of a product

    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.

    Solving a Single Variable Equation:

    Solve  :    x+11 = 0  

     

    Subtract  11  from both sides of the equation :  

                         x = -11

    Solving a Single Variable Equation:

    Solve  :    x+9 = 0  

     

    Subtract  9  from both sides of the equation :  

                         x = -9

    Supplement : Solving Quadratic Equation Directly

    x2+20x+99  = 0   directly  

    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

    Parabola, Finding the Vertex:  

    Find the Vertex of   y = x2+20x+99  

     

    Parabolas 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, 1 , 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  -10.0000

     

    Plugging into the parabola formula  -10.0000  for  x  we can calculate the  y -coordinate :  

    y = 1.0 * -10.00 * -10.00 + 20.0 * -10.00 + 99.0

    or   y = -1.000

    Parabola, Graphing Vertex and X-Intercepts :  

    Root plot for :  y = x2+20x+99  

    Axis of Symmetry (dashed)  {x}={-10.00}  

    Vertex at  {x,y} = {-10.00,-1.00}  

    x -Intercepts (Roots) :  

    Root 1 at  {x,y} = {-11.00, 0.00}  

    Root 2 at  {x,y} = {-9.00, 0.00}

    Solve Quadratic Equation by Completing The Square  

    Solving   x2+20x+99 = 0 by Completing The Square .  

     

    Subtract  99  from both side of the equation :  

      x2+20x = -99  

     

    Now the clever bit: Take the coefficient of  x , which is  20 , divide by two, giving  10 , and finally square it giving  100  

     

    Add  100  to both sides of the equation :  

     On the right hand side we have :  

      -99  +  100    or,  (-99/1)+(100/1)  

    The common denominator of the two fractions is  1   Adding  (-99/1)+(100/1)  gives  1/1  

     So adding to both sides we finally get :  

      x2+20x+100 = 1  

     

    Adding  100  has completed the left hand side into a perfect square :  

      x2+20x+100  =  

      (x+10) • (x+10)  =  

     (x+10)2  

    Things which are equal to the same thing are also equal to one another. Since  

      x2+20x+100 = 1 and  

      x2+20x+100 = (x+10)2  

    then, according to the law of transitivity,  

      (x+10)2 = 1  

     

    We’ll refer to this Equation as  Eq. #3.2.1  

     

    The Square Root Principle says that When two things are equal, their square roots are equal.  

     

    Note that the square root of  

      (x+10)2   is  

      (x+10)2/2 =  

     (x+10)1 =  

      x+10  

     

    Now, applying the Square Root Principle to  Eq. #3.2.1  we get:  

      x+10 = √ 1  

     

    Subtract  10  from both sides to obtain:  

      x = -10 + √ 1  

    Since a square root has two values, one positive and the other negative  

      x2 + 20x + 99 = 0  

    has two solutions:  

      x = -10 + √ 1  

        or  

      x = -10 – √ 1  

    Two solutions were found :  

    1. x = -9  
    2. x = -11

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