- Factor out \(-1\) from the expression: \(-(2h^2 + 15h + 7)\).
- Use the quadratic formula to find the roots of \(2h^2 + 15h + 7 = 0\), which are \(h = -1/2\) and \(h = -7\).
- Factor the expression as \(-(2h+1)(h+7)\).
- Conclude that \((2h+1)\) is a factor of the expression \(-2h^2 - 15h - 7\). The correct option is \((2h+1)\).
Explanation
- Analysis of the Quadratic Expression
We are given the quadratic expression \(-2h^2 - 15h - 7\). To determine its factors, it is helpful to factor out \(-1\), giving us \(-(2h^2 + 15h + 7)\). A linear expression \((ah + b)\) is a factor of the quadratic expression if the root \(h = -b/a\) makes the expression zero. We will find the roots of the equation \(2h^2 + 15h + 7 = 0\) to identify the linear factors.
- Calculating the Roots
To find the roots of the quadratic equation \(2h^2 + 15h + 7 = 0\), we use the quadratic formula \(h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=2, b=15,\) and \(c=7\).
Calculating the discriminant:
\[b^2 - 4ac = 15^2 - 4(2)(7) = 225 - 56 = 169.\]
Since \(\sqrt{169} = 13\), the roots are:
\[h = \frac{-15 \pm 13}{4}.\]
The two roots are:
\[h_1 = \frac{-15 + 13}{4} = \frac{-2}{4} = -\frac{1}{2},\]
\[h_2 = \frac{-15 - 13}{4} = \frac{-28}{4} = -7.\]
- Identifying the Factor and Conclusion
With roots \(h = -1/2\) and \(h = -7\), the expression \(2h^2 + 15h + 7\) can be written in factored form as:
\[2h^2 + 15h + 7 = 2(h + 1/2)(h + 7) = (2h + 1)(h + 7).\]
Thus, the original expression \(-2h^2 - 15h - 7\) is:
\[-(2h^2 + 15h + 7) = -(2h + 1)(h + 7).\]
Comparing this result to the given options \((h+2)\), \((3h-2)\), \((2h+1)\), and \((h-7)\), we see that \((2h+1)\) is a factor of the expression.
Examples
Factoring quadratic expressions is a fundamental skill in algebra with many real-world applications. For example, if you are modeling the trajectory of a ball thrown into the air, its height \(H\) at time \(t\) might be given by a quadratic equation like \(H(t) = -5t^2 + 20t + 5\). By factoring such expressions, you can easily determine the time at which the object hits the ground (when height is zero) or find the peak of its flight, helping engineers and physicists design safer and more efficient systems.