Factor completely.
$5 h^2+24 h-5$
The expression to factor is a quadratic trinomial: \(5h^2 + 24h - 5\). This is of the form \(ah^2 + bh + c\), where \(a = 5\), \(b = 24\), and \(c = -5\). To factor this expression, we aim to express it as the product of two linear binomials by splitting the middle term.
To factor the expression \(5h^2 + 24h - 5\), we look for two integers whose product is \(a \cdot c\) and whose sum is \(b\).
Calculating the product:
\[a \cdot c = 5 \cdot (-5) = -25\]
We need to find two numbers that multiply to \(-25\) and add up to \(24\). By inspecting the factors of \(-25\), we find that \(25\) and \(-1\) satisfy both conditions:
\[25 \cdot (-1) = -25\]
\[25 + (-1) = 24\]
Now we replace the middle term \(24h\) with \(25h - 1h\):
\[5h^2 + 25h - 1h - 5\]
Next, we factor by grouping. We group the first two terms and the last two terms:
\[(5h^2 + 25h) + (-1h - 5)\]
From the first group, we can factor out \(5h\):
\[5h(h + 5)\]
From the second group, we can factor out \(-1\):
\[-1(h + 5)\]
Now, notice that \((h + 5)\) is a common factor:
\[5h(h + 5) - 1(h + 5) = (5h - 1)(h + 5)\]
Thus, the expression \(5h^2 + 24h - 5\) is factored into \((5h - 1)(h + 5)\).
Factoring quadratic expressions is essential in various fields, such as engineering and physics, when solving for variables in equations that describe motion, area, or energy. For example, if the area of a rectangular plot of land is given by \(A = 5h^2 + 24h - 5\) square meters, factoring this expression allows you to determine the length and width of the plot as \((5h - 1)\) and \((h + 5)\) meters, respectively. This simplifies the process of finding the dimensions of the land when you know the overall area.