Question

The quadratic function y = –10x2 + 160x – 430 models a store’s daily profit (y) for selling a T-shirt priced at x dollars. What equation do you need to solve to find the selling price or prices that would generate $50 in daily profit? What method would you use to solve the equation?

Answer 1

Answer:

Replace y in the equation with 50: (50 = -10x2 +160x - 430) Use factoring to solve the equation. After writing the equation in standard form and dividing each side by -10, it is easy to factor as 0 = (x - 4)(x - 12).

Answer 2

 x2-16x+48=0 Two solutions were found : x = 12 x = 4

Step by step solution :Step  1  :Trying to factor by splitting the middle term

 1.1     Factoring  x2-16x+48 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -16x  its coefficient is  -16 .
The last term, "the constant", is  +48 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 48 = 48 

Step-2 : Find two factors of  48  whose sum equals the coefficient of the middle term, which is   -16 .

     -48   +   -1   =   -49     -24   +   -2   =   -26     -16   +   -3   =   -19     -12   +   -4   =   -16   That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -12  and  -4 
                     x2 - 12x - 4x - 48

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-12)
              Add up the last 2 terms, pulling out common factors :
                    4 • (x-12)
Step-5 : Add up the four terms of step 4 :
                    (x-4)  •  (x-12)
             Which is the desired factorization

Equation at the end of step  1  : (x - 4) • (x - 12) = 0 Step  2  :Theory - Roots of a product :

 2.1    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.

Solving a Single Variable Equation :

 2.2      Solve  :    x-4 = 0 

 Add  4  to both sides of the equation : 
                      x = 4 

Solving a Single Variable Equation :

 2.3      Solve  :    x-12 = 0 

 Add  12  to both sides of the equation : 
                      x = 12 

Supplement : Solving Quadratic Equation DirectlySolving  x2-16x+48  = 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 :

 3.1      Find the Vertex of   y = x2-16x+48