Question

Ali and akari were trying to solve the equation: [(x-1)(x-7)=5 ]. Ali said, 'I'll multiply [(x-1)(x-7)] and rewrite the equation as [x²-8x+7=5]. Then I'll subtract [5] from both sides, and use the quadratic formula with [a=1], [b=-8], and [c=2].' Akari said, 'The left-hand side is factored, so I'll use the zero product property.' Whose solution strategy would work?

1) Only Ali's
2) Only Akari's
3) Both
4) Neither

Answer 1

Both Ali and Akari propose valid methods for solving the quadratic equation, which can be solved using Ali's quadratic formula or Akari's zero product property after subtracting 5 from both sides.

Step-by-step explanation:

The question is concerned with solving the quadratic equation [(x-1)(x-7)=5]. Both Ali and Akari propose different methods to solve this equation. Ali suggests expanding the factored form and then applying the quadratic formula to find the value of x, whereas Akari proposes to use the zero product property which can be used directly on factored quadratic equations.

Ali's method is a valid approach, requiring the expansion of the equation to x² - 8x + 7, and then bringing 5 to the left-hand side to get x² - 8x + 2 = 0, which can indeed be solved with the quadratic formula with a=1, b=-8, and c=2. Akari's method is also valid but would require setting the factored equation equal to 5, subtracting 5 from both sides, and then applying the zero product property to the resulting equation (x-1)(x-7)-5=0. While a little more complicated, this method would still yield the correct solution.

Once the roots are obtained using either method, it is important to eliminate terms wherever possible to simplify the algebra and check the answer to see if it is reasonable.