Question

The area of a rectangular garden is given by the quadratic function: A (x) = -6x^2 + 105x– 294

When asked to find the possible length and width of the garden, Joe said the length was - 6x + 21 and
the width was x - 14. Bree answered the same question by saying the length was -3x + 42 and the
width was 2x – 7.
Who is correct and why?

Answer 1

Bree is correct

Explanation:

Given

Area of rectangle;

`A (x) = -6x^2 + 105x - 294`

Joe's Result:

`Length = -6x + 21\\Width = x - 4`

Bree' Result

`Length = -3x + 42\\Width = 2x - 7`

Required

Determine whose result is correct and why.

To determine the correct result, we simply find the roots of the quadratic function

`A (x) = -6x^2 + 105x - 294`

Such that A(x) = 0

`-6x^2 + 105x - 294 = 0`

Start Factorization;

Expand

`A (x) = -6x^2 + 105x - 294`

Group the above expression in 2s

`A(x) = (-6x^2 + 84x) + (21x - 294)`

Factorize

`A(x) =2x(-3x + 42) -7 (-3x + 42)`

`A(x) = (2x - 7)(-3x + 42)`

Recall that; Area of a rectangle is calculated by;

`A(x) = Length * Width`

By comparison;

`Length = 2x - 7\\Width = -3x + 42`

At this point, we can conclude that Bree's computation is correct;

Reason

The product of (-3x + 42) and (2x – 7) will result in the given area of the rectangle