91Ó°ÊÓ

A quadratic equation is a polynomial equation of degree two. Its standard form is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The equation you are solving here is \( 4x^2 + 1 = 0 \), which is already in the standard form because \( a = 4 \), \( b = 0 \), and \( c = 1 \).

Quadratic equations can have:

• Two real solutions,
• One real solution, or
• Two complex solutions.

The nature of the solutions depends on the discriminant, \( b^2 - 4ac \). If the discriminant is negative, as in this example, the solutions will be complex.

The imaginary unit, denoted as \( i \), is a mathematical concept that allows us to work with square roots of negative numbers. By definition, \( i = \sqrt{-1} \). This property is fundamental for handling negative square roots while solving quadratic equations when the discriminant is less than zero.

Since our equation leads to \( x^2 = -\frac{1}{4} \), we naturally encounter the need to utilize the imaginary unit to express the square root of a negative number. The square root property, \( \sqrt{-a^2} = ai \), helps us take these complex solutions. In our example, \( x = \pm \frac{i}{2} \) because the square root of \(-\frac{1}{4}\) is expressed using \( i \).

Remember that any real number can also be expressed in terms of imaginary numbers, specifically, as a complex number where the imaginary part is zero.

Solving equations, specifically quadratic ones, often involves a clear and systematic process. Here’s the step-by-step breakdown for tackling the equation \( 4x^2 + 1 = 0 \):

• Step 1: **Isolate the quadratic term** by rewriting it in standard form. Our starting point was \( 4x^2 + 1 = 0 \).
• Step 2: **Move constant terms** to the other side of the equation. Subtracting 1 gives \( 4x^2 = -1 \).
• Step 3: **Solve for the squared variable** by dividing by 4, leading to \( x^2 = -\frac{1}{4} \).
• Step 4: **Find \( x \) using square roots**, acknowledging the presence of \( i \): \( x = \pm \sqrt{-\frac{1}{4}} \).
• Step 5: **Simplify using the imaginary unit**: Breakdown the square root, resulting in \( x = \pm i \times \frac{1}{2} \).

This logical approach allows us to navigate through quadratic solutions with negative discriminants, ultimately revealing complex solutions.