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Write a polynomial to represent the area of the shaded region. Then solve for x given that the area of the shaded region is 24 square units.
3x2 + x - 6; x = 3
3x2 + 7x - 6; x = 2
3x2 + 7x - 6; x = 5
WILL GIVE BRAINLIEST WITH EXPLANATION

Write a polynomial to represent the area of the shaded region Then solve for x given that the area of the shaded region is 24 square units 3x2 x 6 x 3 3x2 7x 6 class=

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sillyside829

Answer:

[tex]3x^{2}[/tex] +x -6 ; x = 3

Step-by-step explanation:

PART 1: The Polynomial

1. Set up the Equation: (x+3)(3x-2) - 6x

You want to find the area of the larger of the 2 rectangles that way you can subtract the area of the smaller one to get the shaded area (if that makes sense). You multiply length by width to get the area so you will multiply (x+3) by (3x - 2). The same applies to the smaller of the two rectangles, so that will be 6x.

2.FOIL: [tex]3x^{2}[/tex] - 2x + 9x -6 - 6x

You are going to want to foil (x+3) and (3x-2). If you are unfamiliar with the FOIL method I would recommend watching a Khan Academy video about it.

3. Simplify: [tex]3x^{2}[/tex] +x -6

Add and subtract like terms. This will be your polynomial.

PART 2: The Value of x

1. Set up the Equation: [tex]3x^{2}[/tex] +x -6 = 24

You will set the polynomial to equal 24 the polynomial and number both equal the size of the shaded area

2. Subtract: [tex]3x^{2}[/tex] +x -30 = 0

Subtract 24 from both sides.

3. Factor: (3x + 10)(x - 3) = 0

4. Set each group to equal zero.

  a. 3x + 10 = 0

      3x = -10

      x = -10/3

  ----------------

  b. x - 3 = 0

   x = 3

5. Check both the answers in [tex]3x^{2}[/tex] +x -6 = 24 to see if your answers are correct. The answer would be x = 3.

Polynomial representing the area of the shaded region will be (3x² - x - 6) square units and the value of 'x' is 3.

Area of a rectangle:

  • Area of a rectangle is given by the expression,

       Area of a rectangle = Length × Width

Given in the picture,

  • Larger rectangle with measures,

        Length = (x + 3) units

        Width = (3x - 2) units

  • Small rectangle with measures,

        Length = x units

        Width = 6 units

Area of the shaded region = Area of the larger rectangle - Area of the smaller rectangle.

Area of the larger rectangle = (3x - 2) × (x + 3) square units

Area of the smaller rectangle = 6x square units

Area of the shaded region = (3x - 2)(x + 3) - 6x

                                            = 3x(x + 3) - 2(x + 3) - 6x

                                            = 3x² + 9x - 2x - 6 - 6x

                                            = (3x² + x - 6) square units

If the area of the shaded region is 24 square units.

3x² + x - 6 = 24                 [Substitute the values]

3x² + x - 30 = 0

3x² + 10x - 9x - 30 = 0      [Factor the equation]

x(3x + 10) + 3(3x + 10) = 0

(x - 3)(3x + 10) = 0

[tex]x=3,-\frac{10}{3}[/tex]

      Therefore, polynomial representing the area of the shaded region will be (3x² - x - 6) square units and the value of x is 3.

Learn more about the area of a rectangle here,

https://brainly.com/question/1256199?referrer=searchResults

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