91Ó°ÊÓ

Quadratic equations are equations that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. Typically, you can solve them using various methods:

• Factoring
• Using the quadratic formula
• Completing the square

In this context, we focus on using the quadratic formula. This powerful tool allows you to solve any quadratic equation, as long as you know the coefficients \(a\), \(b\), and \(c\). The quadratic formula is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
By substituting the values of \(a\), \(b\), and \(c\) into this formula, you can find the values of \(x\) that make the equation true.
For example, given the quadratic equation \(m^2 + \frac{4}{3}m + \frac{5}{9} = 0\), we quickly identify \(a=1\), \(b=\frac{4}{3}\), and \(c=\frac{5}{9}\), and apply the quadratic formula to find the roots of the equation.

When solving quadratic equations, the nature of the solutions depends heavily on the term inside the square root of the quadratic formula, known as the discriminant. The discriminant is \(b^2 - 4ac\). It determines whether the solutions are real or complex:

• If the discriminant is positive, the equation has two distinct real solutions.
• If the discriminant is zero, the equation has exactly one real solution.
• If the discriminant is negative, the equation has complex (non-real) solutions.

In our specific equation \(m^2 + \frac{4}{3}m + \frac{5}{9} = 0\), the discriminant evaluates to \(-\frac{4}{9}\), which is negative. Therefore, this equation does not have any real solutions. Instead, it has complex solutions. The solutions can be expressed using imaginary numbers when \(\sqrt{a negative number}\) is taken into account, yielding solutions like \(- \frac{4}{3} \pm \frac{2i}{3}\) for this quadratic. Understanding whether solutions are real or complex is crucial as it determines the nature of the roots and their applicability in real-world situations.

Sometimes, quadratic equations contain fractional coefficients, as seen in \(m^2 + \frac{4}{3}m + \frac{5}{9} = 0\). Handling these may seem intimidating initially, but the procedure is quite similar to equations with integer coefficients.

• First, identify \(a\), \(b\), and \(c\). In this case, \(a=1\), \(b=\frac{4}{3}\), and \(c=\frac{5}{9}\).
• Plug into the quadratic formula \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Calculate the discriminant and proceed to determine the solutions as real or complex.

When fractions are involved, be extra careful with arithmetic calculations to prevent errors. Convert them to a common denominator if possible, to simplify the computation. Despite the added complexity, solving quadratic equations with fractional coefficients follows the same underlying principles as those with whole numbers, making them a task that becomes easier with practice.