factored form of y=(x-1)^2-4

Question

factored form of y=(x-1)^2-4

 

also standard form of the same equation 

Answer 1

     y = (x - 1)² - 4 

Notice that both terms on the right hand side of the equation are perfect squares, and a binomial of this form is known as a difference of two squares. That is, 

         a binomial of the form   a² - b²   , has as its factors   a + b   and   a - b

         ==>   a² - b² = (a + b)(a - b)

   y = (x - 1)² - 4     ==>     y = (x - 1)² - (2)²    

                             ==>     y = ((x - 1) + 2)((x - 1) - 2)

                             ==>     y = (x - 1 + 2)(x - 1 - 2)

                             ==>     y = (x + 1)(x - 3)     (this is the factored form)

The standard form of a quadratic equation, like this one, is written as follows:

          y = ax² + bx + c

   y = (x - 1)² - 4     ==>     y = (x - 1)(x - 1) - 4

                                         y = (x² - 1x - 1x + 1) - 4

                                         y = x² - 2x + 1 - 4

                                         y = x² - 2x - 3             (this is the standard form)

You can also use the factored form we found above to find the standard form:

        y = (x - 1)² - 4     ==>     y = (x + 1)(x - 3)

                                              y = x·x + x·-3 + 1·x + 1·-3

                                              y = x² - 3x + x - 3

                                              y = x² - 2x - 3