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The Quadratic Formula is a crucial tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). In our exercise, after rearranging, the quadratic equation becomes \( 2x^2 - x + 1 = 0 \). Using the values of \( a = 2 \), \( b = -1 \), and \( c = 1 \), you can apply the quadratic formula, given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps you find the roots of any quadratic equation with ease. You simply plug in the values of \( a \), \( b \), and \( c \), then simplify the expression under the square root, known as the discriminant \( b^2 - 4ac \). The \( \pm \) symbol indicates that there are generally two solutions, known as the roots of the equation. These roots can be real or complex depending on the value under the square root.

In mathematics, complex numbers extend the concept of real numbers to include values that are not on the number line. They are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. A complex number consists of two parts:

• The real part \( a \)
• The imaginary part \( bi \)

In our exercise, when we apply the quadratic formula, we end up with a negative number under the square root, \( \sqrt{-7} \), which indicates the presence of complex roots. This means our solutions include imaginary parts, thus both solutions of the quadratic involve complex numbers. The roots are expressed as \( \frac{1 \pm i \sqrt{7}}{4} \), showing how both real and imaginary components combine in the answer.

Imaginary numbers are crucial for understanding complex numbers. They arise when we try to take the square root of a negative number, a task that's impossible with real numbers alone. To accommodate this, mathematicians use \( i \), the square root of \( -1 \). With \( i \), any negative number can be expressed in terms of a real number and an imaginary coefficient, figured by multiplying with \( i \).
In the quadratic formula exercise, \( \sqrt{-7} \) becomes \( i \sqrt{7} \). This conversion allows for the expression of roots in terms of complex numbers. Consequently, our solutions of the equation become \( \frac{1 + i \sqrt{7}}{4} \) and \( \frac{1 - i \sqrt{7}}{4} \). These solutions highlight how imaginary numbers work in tandem with real numbers to form complex solutions. Understanding imaginary numbers enables a deeper grasp of many complex problems in mathematics.