- Identify coefficients \(a=1, b=16, c=61\) for the equation \(n^2+16n+61=0\).
- Apply the quadratic formula \(n = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find the roots \(n = -8 \pm \sqrt{3}\).
- Generalize the quadratic formula for \(Ak^2+Bk+C=0\) as \(k = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\).
- Explain that when the discriminant is negative, use \(i=\sqrt{-1}\) to express complex roots as \(k = \frac{-B \pm i\sqrt{|B^2-4AC|}}{2A}\).
- Final answers: \(n = \boxed{-8 + \sqrt{3}}, n = \boxed{-8 - \sqrt{3}}\).
Explanation
- Analyzing the Quadratic Equation and Identifying Coefficients
To solve the quadratic equation \(n^2 + 16n + 61 = 0\), we first identify the coefficients in the general form \(an^2 + bn + c = 0\). Here, \(a = 1\), \(b = 16\), and \(c = 61\). The quadratic formula, which provides the solutions for any such equation, is given by:
\[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula is fundamental for finding the roots of quadratic equations, whether they are real or complex. We will use this to determine the values of \(n\).
- Calculating the Roots for \(n\).
We substitute the coefficients \(a = 1\), \(b = 16\), and \(c = 61\) into the quadratic formula:
\[n = \frac{-16 \pm \sqrt{16^2 - 4(1)(61)}}{2(1)}\]
First, we calculate the discriminant (\(b^2 - 4ac\)):
\[16^2 = 256\]
\[4 \times 1 \times 61 = 244\]
\[16^2 - 4(1)(61) = 256 - 244 = 12\]
Now we substitute the discriminant back into the formula:
\[n = \frac{-16 \pm \sqrt{12}}{2}\]
Since \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\), the expression simplifies to:
\[n = \frac{-16 \pm 2\sqrt{3}}{2} = -8 \pm \sqrt{3}\]
Thus, the solutions are \(n = -8 + \sqrt{3}\) and \(n = -8 - \sqrt{3}\).
- Procedure for Complex Solutions of \(k\).
For a general quadratic equation \(Ak^2 + Bk + C = 0\), the solutions for \(k\) are found using the quadratic formula:
\[k = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\]
When the discriminant \(D = B^2 - 4AC\) is negative, the square root \(\sqrt{D}\) involves an imaginary component. We use the imaginary unit \(i\), defined by \(i = \sqrt{-1}\), to express these roots. Specifically, if \(D < 0\), we can write \(\sqrt{D} = \sqrt{-1 \times |D|} = i\sqrt{|D|}\).
The roots are then:
\[k = \frac{-B \pm i\sqrt{|B^2 - 4AC|}}{2A}\]
This process allows us to find complex roots for any quadratic equation that does not have real solutions.
Examples
Quadratic equations are essential in many real-life applications. For instance, if you are designing a trajectory for a projectile, the height \(h\) at time \(t\) can be modeled by a quadratic equation like \(h(t) = -gt^2 + v_0t + h_0\). Solving for \(h(t) = 0\) helps determine when the projectile hits the ground. Similarly, engineers use these equations to optimize the dimensions of rectangular structures, ensuring the area or perimeter meets specific requirements, which is a foundational skill in physics and architecture.