Question

what is the width of x(2x+3)=560

Answer 1

To find the width in the equation \( x(2x + 3) = 560 \), we first need to rearrange it into a standard quadratic form:

\[
2x^2 + 3x - 560 = 0
\]

Next, we can apply the quadratic formula, which is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our equation \( 2x^2 + 3x - 560 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = 3 \)
- \( c = -560 \)

Now we need to calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 - 4ac = 3^2 - 4 \cdot 2 \cdot (-560) = 9 + 4480 = 4489
\]

Now we take the square root of the discriminant:

\[
\sqrt{4489} = 67
\]

Now we can use the quadratic formula:

\[
x = \frac{-3 \pm 67}{2 \cdot 2}
\]

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

1. \( x = \frac{-3 + 67}{4} = \frac{64}{4} = 16 \)
2. \( x = \frac{-3 - 67}{4} = \frac{-70}{4} = -17.5 \)

Since width cannot be negative, we take the positive value:

\[
x = 16
\]

Thus, the width \( x \) is \( 16 \).