- Transform the equation \(x^2 - 6x = -58\) to standard form: \(x^2 - 6x + 58 = 0\).
- Apply the quadratic formula \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\) with \(A=1, B=-6, C=58\).
- Solve for \(x\) to get \(x = 3 \pm 7i\).
- The solutions are \(\boxed{3 \pm 7i}\).
Explanation
- Rewrite the equation in standard form and identify coefficients
To solve the quadratic equation \(x^2 - 6x = -58\), we first write it in the standard quadratic form \(Ax^2 + Bx + C = 0\). Adding 58 to both sides, we get:
\[x^2 - 6x + 58 = 0.\]
Here, \(A = 1\), \(B = -6\), and \(C = 58\). We can solve this equation using the quadratic formula:
\[x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}.\]
- Apply the quadratic formula
Substitute the values \(A = 1\), \(B = -6\), and \(C = 58\) into the quadratic formula:
\[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(58)}}{2(1)}.\]
Calculating the discriminant:
\[(-6)^2 - 4(1)(58) = 36 - 232 = -196.\]
Now, substitute this back into the formula for \(x\):
\[x = \frac{6 \pm \sqrt{-196}}{2}.\]
- Find the solutions and identify \(a\) and \(b\)
Since the discriminant is negative, the solutions involve the imaginary unit \(i\), where \(i^2 = -1\). We know that \(\sqrt{-196} = \sqrt{196} \cdot \sqrt{-1} = 14i\). Thus:
\[x = \frac{6 \pm 14i}{2}.\]
Dividing both terms in the numerator by 2, we find:
\[x = 3 \pm 7i.\]
This gives the two complex solutions \(x = 3 + 7i\) and \(x = 3 - 7i\). Therefore, we identify \(a = 3\) and \(b = 7\).
Examples
Quadratic equations with complex roots are essential in fields like electrical engineering, where they describe the impedance in alternating current (AC) circuits. The real part (\(a\)) and imaginary part (\(b\)) of the complex solution often correspond to resistance and reactance, respectively. Understanding how to find these values allows engineers to analyze circuit stability and performance efficiently.