Question

what is the width of what is the length of 4x(2x+1)=220

Answer 1

To find the width and length in the equation \( 4x(2x + 1) = 220 \), we first need to solve for \( x \).

1. Start by simplifying the equation:

\[
4x(2x + 1) = 220
\]

Expanding the left side gives:

\[
8x^2 + 4x = 220
\]

2. Move all terms to one side to set the equation to zero:

\[
8x^2 + 4x - 220 = 0
\]

3. Simplify the equation by dividing all terms by 4:

\[
2x^2 + x - 55 = 0
\]

4. Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 2 \), \( b = 1 \), and \( c = -55 \):

\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-55)}}{2 \cdot 2}
\]

Calculate the discriminant:

\[
1 + 440 = 441
\]

Now find the square root:

\[
\sqrt{441} = 21
\]

5. Substitute back into the quadratic formula:

\[
x = \frac{-1 \pm 21}{4}
\]

Calculating the two possible values for \( x \):

- For \( x = \frac{20}{4} = 5 \)
- For \( x = \frac{-22}{4} = -5.5 \) (not valid since dimensions cannot be negative)

6. Therefore, the valid value for \( x \) is \( 5 \).

Now, if we interpret \( 4x \) as the length and \( (2x + 1) \) as the width:

- Length:

\[
4x = 4(5) = 20
\]

- Width:

\[
2x + 1 = 2(5) + 1 = 10 + 1 = 11
\]

Thus, the length is \( 20 \) and the width is \( 11 \).