Question

what is the height of 1/2 x(x+8)=154

Answer 1

To solve the equation \(\frac{1}{2} x(x + 8) = 154\), we first eliminate the fraction by multiplying both sides by 2:

\[
x(x + 8) = 308
\]

Next, we expand the left-hand side:

\[
x^2 + 8x = 308
\]

Now, we can rearrange the equation into standard quadratic form:

\[
x^2 + 8x - 308 = 0
\]

Next, we will use the quadratic formula to find the roots of the quadratic equation. The quadratic formula is:

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

Here, \(a = 1\), \(b = 8\), and \(c = -308\). We first calculate the discriminant:

\[
b^2 - 4ac = 8^2 - 4(1)(-308) = 64 + 1232 = 1296
\]

Now we substitute back into the quadratic formula:

\[
x = \frac{-8 \pm \sqrt{1296}}{2(1)}
\]

Calculating \(\sqrt{1296}\):

\[
\sqrt{1296} = 36
\]

Now, substituting back in gives us:

\[
x = \frac{-8 \pm 36}{2}
\]

This results in two possible solutions:

1.

\[
x = \frac{-8 + 36}{2} = \frac{28}{2} = 14
\]

2.

\[
x = \frac{-8 - 36}{2} = \frac{-44}{2} = -22
\]

Since height cannot be negative, the valid solution is:

\[
x = 14
\]

Thus, the height is \(14\).