15x 2 7x 4 90quadratic

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.

Root plot for :  y = 15x2-7x-4 Axis of Symmetry (dashed)  {x}={ 0.23}  Vertex at  {x,y} = { 0.23,-4.82}   x -Intercepts (Roots) : Root 1 at  {x,y} = {-0.33, 0.00}  Root 2 at  {x,y} = { 0.80, 0.00}

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

2.1     Factoring  15x2-7x-4  The first term is,  15x2  its coefficient is  15 .The middle term is,  -7x  its coefficient is  -7 .The last term, "the constant", is  -4 Step-1 : Multiply the coefficient of the first term by the constant   15 • -4 = -60 Step-2 : Find two factors of  -60  whose sum equals the coefficient of the middle term, which is   -7 .

15x 2 7x 4 90calculator

Forming taps must be applied in materials that cold form well. This includes light metals and light metal alloys as well as steels and other materials with tensile strength to 1200 N/mm2 and hardness below about RC 35 – 40. Generally, materials that produce a continuous chip when drilling are good candidates for thread forming. This list is longer than you might think, and includes wrought and cast aluminum and aluminum alloys, copper, brass, stainless steels, carbon steels, and zinc diecasting alloys.

3.3      Solve  :    3x+1 = 0  Subtract  1  from both sides of the equation :                       3x = -1 Divide both sides of the equation by 3:                     x = -1/3 = -0.333

15x 2 7x 4 90answer

4.3     Solving    15x2-7x-4 = 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   =   -4 Accordingly,  B2  -  4AC   =                     49 - (-240) =                      289Applying the quadratic formula :                7 ± √ 289    x  =    —————                    30Can  √ 289 be simplified ?Yes!   The prime factorization of  289   is   17•17  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).√ 289   =  √ 17•17   =                ±  17 • √ 1   =                ±  17 So now we are looking at:           x  =  ( 7 ± 17) / 30Two real solutions:x =(7+√289)/30=(7+17)/30= 0.800 or:x =(7-√289)/30=(7-17)/30= -0.333

15x 2 7x 4 90graph

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -12  and  5                      15x2 - 12x + 5x - 4Step-4 : Add up the first 2 terms, pulling out like factors :                    3x • (5x-4)              Add up the last 2 terms, pulling out common factors :                     1 • (5x-4) Step-5 : Add up the four terms of step 4 :                    (3x+1)  •  (5x-4)             Which is the desired factorization

But many of those threads could be produced using forming taps, which can provide multiple advantages. For ductile work materials, thread forming can provide better size control and stronger threads while improving tool life and productivity.

4.1      Find the Vertex of   y = 15x2-7x-4Parabolas 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 - 4.0 or   y = -4.817

4.2     Solving   15x2-7x-4 = 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-(4/15) = 0Add  4/15  to both side of the equation :    x2-(7/15)x = 4/15Now 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 :   4/15  +  49/900   The common denominator of the two fractions is  900   Adding  (240/900)+(49/900)  gives  289/900   So adding to both sides we finally get :   x2-(7/15)x+(49/900) = 289/900Adding  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) = 289/900 and   x2-(7/15)x+(49/900) = (x-(7/30))2 then, according to the law of transitivity,   (x-(7/30))2 = 289/900We'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-(7/30))2   is   (x-(7/30))2/2 =  (x-(7/30))1 =   x-(7/30)Now, applying the Square Root Principle to  Eq. #4.2.1  we get:   x-(7/30) = √ 289/900 Add  7/30  to both sides to obtain:   x = 7/30 + √ 289/900 Since a square root has two values, one positive and the other negative   x2 - (7/15)x - (4/15) = 0   has two solutions:  x = 7/30 + √ 289/900    or  x = 7/30 - √ 289/900 Note that  √ 289/900 can be written as  √ 289  / √ 900   which is 17 / 30

The vast majority of threaded holes–more than 90%, according to one supplier of taps and other cutting tools–are produced using cutting taps.

Unlike thread cutting, no material is removed during thread forming. Rather, the process displaces the material to generate the thread form. Since the metal’s structure is cold worked along the thread profile, the threads produced are generally stronger and have a smooth, burnished surface finish.