15x 2 7x 4 90graph

3.2     Solving   15x2+7x-42 = 0 by Completing The Square . Divide both sides of the equation by  15  to have 1 as the coefficient of the first term :   x2+(7/15)x-(14/5) = 0Add  14/5  to both side of the equation :    x2+(7/15)x = 14/5Now the clever bit: Take the coefficient of  x , which is  7/15 , divide by two, giving  7/30 , and finally square it giving  49/900 Add  49/900  to both sides of the equation :  On the right hand side we have :   14/5  +  49/900   The common denominator of the two fractions is  900   Adding  (2520/900)+(49/900)  gives  2569/900   So adding to both sides we finally get :   x2+(7/15)x+(49/900) = 2569/900Adding  49/900  has completed the left hand side into a perfect square :   x2+(7/15)x+(49/900)  =   (x+(7/30)) • (x+(7/30))  =  (x+(7/30))2 Things which are equal to the same thing are also equal to one another. Since   x2+(7/15)x+(49/900) = 2569/900 and   x2+(7/15)x+(49/900) = (x+(7/30))2 then, according to the law of transitivity,   (x+(7/30))2 = 2569/900We'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+(7/30))2   is   (x+(7/30))2/2 =  (x+(7/30))1 =   x+(7/30)Now, applying the Square Root Principle to  Eq. #3.2.1  we get:   x+(7/30) = √ 2569/900 Subtract  7/30  from both sides to obtain:   x = -7/30 + √ 2569/900 Since a square root has two values, one positive and the other negative   x2 + (7/15)x - (14/5) = 0   has two solutions:  x = -7/30 + √ 2569/900    or  x = -7/30 - √ 2569/900 Note that  √ 2569/900 can be written as  √ 2569  / √ 900   which is √ 2569  / 30

15x 2 7x 4 90quadratic

3.3     Solving    15x2+7x-42 = 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   =     15                      B   =    7                      C   =  -42 Accordingly,  B2  -  4AC   =                     49 - (-2520) =                      2569Applying the quadratic formula :                -7 ± √ 2569    x  =    ——————                      30  √ 2569   , rounded to 4 decimal digits, is  50.6853 So now we are looking at:           x  =  ( -7 ±  50.685 ) / 30Two real solutions: x =(-7+√2569)/30= 1.456 or: x =(-7-√2569)/30=-1.923

For tidiness, printing of 18 lines which failed to find two such factors, was suppressedObservation : No two such factors can be found !! Conclusion : Trinomial can not be factored

15x 2 7x 4 90calculator

Root plot for :  y = 15x2+7x-42 Axis of Symmetry (dashed)  {x}={-0.23}  Vertex at  {x,y} = {-0.23,-42.82}   x -Intercepts (Roots) : Root 1 at  {x,y} = {-1.92, 0.00}  Root 2 at  {x,y} = { 1.46, 0.00}

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2.1     Factoring  15x2+7x-42  The first term is,  15x2  its coefficient is  15 .The middle term is,  +7x  its coefficient is  7 .The last term, "the constant", is  -42 Step-1 : Multiply the coefficient of the first term by the constant   15 • -42 = -630 Step-2 : Find two factors of  -630  whose sum equals the coefficient of the middle term, which is   7 .

15x 2 7x 4 90answer

3.1      Find the Vertex of   y = 15x2+7x-42Parabolas 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, 15 , 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  -0.2333   Plugging into the parabola formula  -0.2333  for  x  we can calculate the  y -coordinate :   y = 15.0 * -0.23 * -0.23 + 7.0 * -0.23 - 42.0 or   y = -42.817