- Rearrange the equation into standard form: \(x^2 - 10x + 34 = 0\).
- Identify coefficients: \(a=1, b=-10, c=34\).
- Calculate the discriminant: \(\Delta = b^2 - 4ac = -36\).
- Apply the quadratic formula to find the complex roots: \(x = \frac{10 \pm 6i}{2} = 5 \pm 3i\).
- The correct option is \(\boxed{x = 5 \pm 3i}\).
Explanation
- Problem Analysis
The objective is to find the solutions to the quadratic equation \(x^2 - 10x = -34\). A quadratic equation is an equation of the second degree, typically written in the standard form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a
eq 0\). Our first task is to rewrite the given equation into this standard form.
- Standard Form and Coefficients
To bring the equation \(x^2 - 10x = -34\) into the standard form \(ax^2 + bx + c = 0\), we add \(34\) to both sides of the equation:
\[x^2 - 10x + 34 = 0\]
Now, we identify the coefficients:
\(a = 1\)
\(b = -10\)
\(c = 34\)
We will use the quadratic formula to find the roots of this equation:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- Solving the Equation
First, we calculate the discriminant, \(\Delta = b^2 - 4ac\), which determines the nature of the roots:
\[\Delta = (-10)^2 - 4(1)(34)\]
\[\Delta = 100 - 136 = -36\]
Since the discriminant is negative (\(\Delta < 0\)), the equation will have two complex conjugate solutions. We use the imaginary unit \(i\), where \(i^2 = -1\) and \(\sqrt{-36} = \sqrt{36} \times \sqrt{-1} = 6i\).
Substituting these into the quadratic formula:
\[x = \frac{-(-10) \pm \sqrt{-36}}{2(1)}\]
\[x = \frac{10 \pm 6i}{2}\]
Simplifying the expression by dividing both terms in the numerator by 2, we obtain:
\[x = 5 \pm 3i\]
- Final Conclusion
Comparing our result \(x = 5 \pm 3i\) with the given options:
- \(x = -8, -2\)
- \(x = 5 \pm 3i\)
- \(x = -5 \pm 3i\)
- \(x = -5 \pm \sqrt{59}\)
The second option is the correct solution. Thus, the solutions to the equation \(x^2 - 10x = -34\) are \(x = 5 \pm 3i\). Therefore, the correct choice is \(\boxed{x = 5 \pm 3i}\).
Examples
Quadratic equations frequently arise in physics and engineering, particularly when analyzing vibrations or alternating current (AC) circuits. For example, when studying the dampening of an oscillator or calculating the impedance in a circuit, one often encounters characteristic equations whose solutions indicate the system's stability or resonant frequency. Being able to solve these equations even when they yield complex roots is essential for predicting the system's behavior over time.