Question

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

Answer 1

To find the value of \( x \) in the equation \( x(2x + 3) = 560 \), we can first expand the equation:

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

Now, we can solve this quadratic equation using the quadratic formula, which is given by:

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

In our equation, \( a = 2 \), \( b = 3 \), and \( c = -560 \). Now we will substitute these values into the quadratic formula.

First, we need to compute the discriminant (\( b^2 - 4ac \)):

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

Now we calculate \( \sqrt{4489} \):

\[
\sqrt{4489} = 67
\]

Now substituting into the quadratic formula gives us:

\[
x = \frac{-3 \pm 67}{2 \cdot 2} = \frac{-3 \pm 67}{4}
\]

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

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

Since \( x \) represents a length, we only take the positive solution:

\[
x = 16
\]

Therefore, the length \( x \) is \( 16 \).