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The quadratic formula is a reliable tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula helps us find solutions when factoring is difficult or impossible. The quadratic formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula involves:

• Identifying the coefficients \( a \), \( b \), and \( c \) from the equation.
• Plugging these values into the formula.
• Calculating the discriminant \( b^2 - 4ac \), an integral part of the formula.

For the quadratic equation \( \frac{1}{2}x^2 + 3x + \frac{13}{2} = 0 \), the quadratic formula provides a straightforward path to find solutions without trial and error. By carefully substituting the known values, you can determine both real and complex solutions.

Complex solutions arise when the discriminant in the quadratic formula is negative. A negative discriminant indicates that there are no real-number solutions to the equation. Instead, the solutions involve imaginary numbers, represented by \( i \), where \( i = \sqrt{-1} \).In our example, with a discriminant of \(-4\), substituting this into the quadratic formula gives:\[x = \frac{-3 \pm \sqrt{-4}}{2 \cdot \frac{1}{2}} \]Since \( \sqrt{-4} = 2i \), the equation simplifies further to:\[x = -3 \pm i\]Thus, the complex solutions are \(-3 + i\) and \(-3 - i\). Complex numbers express solutions as a sum of a real part and an imaginary part, expanding your understanding beyond the real number system.

The discriminant \( b^2 - 4ac \) in algebra plays a crucial role in determining the nature of the solutions for a quadratic equation. The value of the discriminant tells us whether the solutions are real or complex:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is one real solution (a repeated root).
• If the discriminant is negative, there are two complex conjugate solutions.

For the equation \( \frac{1}{2}x^2 + 3x + \frac{13}{2} = 0 \), the discriminant calculated was \(-4\). This negative value is a clear signal that the equation does not have real solutions. Instead, it results in complex solutions, as explored. Understanding the discriminant helps predict solution types quickly, making it an invaluable tool in algebra.