- Rearrange the equation to the standard form: \(2x^2 + 3x + 5 = 0\).
- Determine the discriminant: \(D = 3^2 - 4(2)(5) = -31\).
- Apply the quadratic formula: \(x = \frac{-3 \pm \sqrt{-31}}{4}\).
- Simplify to identify the final complex solutions: \(x = \boxed{\frac{-3 \pm i\sqrt{31}}{4}}\).
Explanation
- Rearranging the equation and identifying coefficients
The given equation is \(2x^2 + 3x = -5\). To solve this quadratic equation, we first bring it into the standard form \(ax^2 + bx + c = 0\). By adding 5 to both sides, we obtain:
\[2x^2 + 3x + 5 = 0\]
From this standard form, we identify the coefficients:
\(a = 2\)
\(b = 3\)
\(c = 5\)
- Calculating the discriminant
To solve the quadratic equation, we calculate the discriminant, denoted by \(D\), which is given by the formula \(D = b^2 - 4ac\). Substituting our values for \(a\), \(b\), and \(c\):
\[D = (3)^2 - 4(2)(5)\]
\[D = 9 - 40\]
\[D = -31\]
Since the discriminant is negative, the equation has complex roots.
- Applying the quadratic formula
The quadratic formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substituting the values \(a = 2\), \(b = 3\), and \(D = -31\) into the formula:
\[x = \frac{-3 \pm \sqrt{-31}}{2(2)}\]
Using the property of complex numbers where \(\sqrt{-1} = i\), we can write \(\sqrt{-31}\) as \(i\sqrt{31}\):
\[x = \frac{-3 \pm i\sqrt{31}}{4}\]
- Determining the final solution
Comparing our result \(x = \frac{-3 \pm i\sqrt{31}}{4}\) with the given options:
- \(\frac{-3 \pm i\sqrt{31}}{4}\)
- \(\frac{-3 \pm i\sqrt{31}}{2}\)
- \(\frac{3 \pm i\sqrt{31}}{4}\)
- \(\frac{-3 \pm i\sqrt{50}}{4}\)
The first option matches our calculated result exactly. Therefore, the solution to the equation \(2x^2 + 3x = -5\) is \(\frac{-3 \pm i\sqrt{31}}{4}\).
Examples
Quadratic equations with complex roots are frequently encountered in electrical engineering and physics when modeling oscillatory systems. For instance, when analyzing an RLC circuit (a circuit with a resistor, inductor, and capacitor), the characteristic equation often results in a quadratic form with a negative discriminant. Solving these equations helps engineers determine the damping behavior of the circuit, ensuring that systems like audio speakers or radio receivers function correctly without unwanted oscillations.