A rectangle has side lengths of (11-x) m and (x-4) m. The area of the rectangle is 15m2. Which quadratic equation in standard form represents the area of this rectangle?

Step-by-step explanation:

the area is : (1-x) (x-4) m² and (1 - x > 0 and x - 4 > 0)

(1-x) (x-4) = x - 4 - x²+4x

(1-x) (x-4) = - x² + 5x - 4 ... quadratic equation in standard form

Area of the rectangle = - (x - 15/2) ² + 49/4

Step-by-step explanation:

* Lets revise the rule of the area of the rectangle

- Any rectangle has two dimensions, length and width

-The area of the rectangle = length * width

* In our problem the dimensions of the rectangle are

(11 - x) and (x - 4)

∴ The area of the rectangle = (11 - x) (x - 4)

- Lets multiply the two brackets

∴ (11 - x) (x - 4) = 11 * x + 11 * (-4) + (-x) * (x) + (-x) * (-4)

∴ (11 - x) (x - 4) = 11x - 44 - x² + 4x ⇒ collect the like terms

∴ (11 - x) (x - 4) = - x² + 15x - 44

* The area of the rectangle = - x² + 15x - 44

- To put them in the standard form a (x + h) ² + k, lets equate

the two forms to find a, h and k

∵ - x² + 15x - 44 = a (x + h) ² + k

∴ - x² + 15x - 44 = a (x² + 2hx + h²) + k

∴ - x² + 15x - 44 = ax² + 2ahx + ah² + k

* Compare the two sides

∵ - 1 = a ⇒ coefficient of x²

∴ a = - 1

∵ 2ah = 15 ⇒ coefficient of x

∴ 2 (-1) h = 15 ⇒ - 2h = 15 ⇒ divide the both sides by - 2

∴ h = - 15/2 = - 7.5

∵ ah² + k = - 44

∴ (-1) (-15/2) ² + k = - 44

∴ - 225/4 + k = - 44 ⇒ add - 225/4 to the both sides

∴ k = 49/4 = 12.25

* The standard form of the equation is - (x - 15/2) ² + 49/4

∴ The quadratic equation represented the area is

A = - (x - 15/2) ² + 49/4