- Identify coefficients \(a=1, b=-5, c=-4\) from the quadratic equation \(x^2 - 5x - 4 = 0\).
- Calculate the discriminant \(D = (-5)^2 - 4(1)(-4) = 41\).
- Use the quadratic formula \(x = \frac{5 \pm \sqrt{41}}{2}\) to find the two roots: \(x_1 \approx 5.7\) and \(x_2 \approx -0.7\).
- Conclude the negative solution is \(\boxed{-0.7}\).
Explanation
- Analyzing the Equation and Applying the Quadratic Formula
To solve the quadratic equation \(0 = x^2 - 5x - 4\), we use the quadratic formula, which states that the solutions for an equation of the form \(ax^2 + bx + c = 0\) are given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In this specific equation, the coefficients are \(a = 1\), \(b = -5\), and \(c = -4\). We substitute these values into the formula to find the roots.
- Calculating the Roots
First, we calculate the discriminant \(D = b^2 - 4ac\):
\[D = (-5)^2 - 4(1)(-4) = 25 + 16 = 41\]
Next, we substitute the values of \(a, b,\) and \(D\) into the quadratic formula to find the roots:
\[x = \frac{-(-5) \pm \sqrt{41}}{2(1)} = \frac{5 \pm \sqrt{41}}{2}\]
Calculating the approximate value of \(\sqrt{41}\), which is approximately \(6.403\), we find the two potential solutions:
\[x_1 = \frac{5 + 6.403}{2} \approx \frac{11.403}{2} \approx 5.7015\]
\[x_2 = \frac{5 - 6.403}{2} \approx \frac{-1.403}{2} \approx -0.7015\]
- Identifying the Negative Solution
We are asked to identify the negative real solution from the provided options. Comparing our calculated values \(5.7015\) and \(-0.7015\) with the given choices, we see that the negative root is approximately \(-0.7015\). Rounding to the nearest tenth, we get \(-0.7\). Therefore, the negative solution is \(-0.7\).
Examples
Quadratic equations appear frequently in engineering and physics, such as when calculating the trajectory of an object thrown into the air. If you define a coordinate system where ground level is at a height of 0, the equation for height over time might be a quadratic. Finding the roots of that equation tells you exactly when the object hits the ground, allowing you to predict its flight duration and landing time. This skill is essential for everything from sports analytics to designing safety systems for moving vehicles.