Question

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

Answer 1

To solve the equation \( 4x(2x + 1) = 220 \), we can first simplify and rearrange it.

1. Distribute \( 4x \) on the left side:
\[
8x^2 + 4x = 220
\]

2. Rearrange the equation to set it to zero:
\[
8x^2 + 4x - 220 = 0
\]

3. To simplify, we can divide the entire equation by 4:
\[
2x^2 + x - 55 = 0
\]

Next, we can solve this quadratic equation using the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \), where \( a = 2 \), \( b = 1 \), and \( c = -55 \).

4. Calculate the discriminant:
\[
b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot (-55) = 1 + 440 = 441
\]

5. Now, use the quadratic formula:
\[
x = \frac{{-1 \pm \sqrt{441}}}{2 \cdot 2}
\]
\[
= \frac{{-1 \pm 21}}{4}
\]

This gives us two potential solutions for \( x \):
- For the positive root:
\[
x = \frac{{20}}{4} = 5
\]
- For the negative root:
\[
x = \frac{{-22}}{4} = -5.5
\]

Since \( x \) represents a width or length in this context, we take the positive solution. Thus, the length \( x \) is:

\[
\boxed{5}
\]