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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?

Respuesta :

MrRoyal

Answer:

Bree is correct

Step-by-step explanation:

Given

Area of rectangle;

[tex]A (x) = -6x^2 + 105x - 294[/tex]

Joe's Result:

[tex]Length = -6x + 21\\Width = x - 4[/tex]

Bree' Result

[tex]Length = -3x + 42\\Width = 2x - 7[/tex]

Required

Determine whose result is correct and why.

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

[tex]A (x) = -6x^2 + 105x - 294[/tex]

Such that A(x) = 0

[tex]-6x^2 + 105x - 294 = 0[/tex]

Start Factorization;

Expand

[tex]A (x) = -6x^2 + 105x - 294[/tex]

Group the above expression in 2s

[tex]A(x) = (-6x^2 + 84x) + (21x - 294)[/tex]

Factorize

[tex]A(x) =2x(-3x + 42) -7 (-3x + 42)[/tex]

[tex]A(x) = (2x - 7)(-3x + 42)[/tex]

Recall that; Area of a rectangle is calculated by;

[tex]A(x) = Length * Width[/tex]

By comparison;

[tex]Length = 2x - 7\\Width = -3x + 42[/tex]

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