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A quadratic equation is a type of polynomial equation that involves the square of the unknown variable. Generally, it is expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The highest degree of the variable is 2, meaning it involves terms like \( x^2 \). Quadratic equations are fundamental in algebra and appear in various applications, from physics to finance, because they can describe parabola-shaped graphs and motions. Breaking down the given example, \( x^2 - 3x + 3 = 0 \), we identify:

• a (the coefficient of \( x^2 \)) is 1
• b (the coefficient of \( x \)) is -3
• c (the constant term) is 3

Solving such equations typically involves finding the values of \( x \) that make the equation true, also known as the solutions or roots.

The discriminant is an important component in solving quadratic equations and is denoted by the symbol \( D \). The formula for the discriminant is given by \( D = b^2 - 4ac \). This value tells us about the nature of the roots of the quadratic equation, whether they are real or complex numbers.
In the example \( x^2 - 3x + 3 = 0 \):

• \( b = -3 \) and \( a = 1 \), \( c = 3 \).
• Calculate \( D \): \( D = (-3)^2 - 4 \cdot 1 \cdot 3 = 9 - 12 = -3 \).

A negative discriminant (

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is exactly one real root (or a repeated root).
• If \( D < 0 \), there are no real roots; instead, there are two complex conjugate roots.

In our equation, \( D = -3 \), which means the equation has two complex conjugate roots.

The quadratic formula provides a method for solving quadratic equations and finding their roots. It is a universal solution that can solve any quadratic equation when you know the coefficients \( a \), \( b \), and \( c \). The quadratic formula is expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In the provided equation \( x^2 - 3x + 3 = 0 \):

• \( a = 1 \), \( b = -3 \), and \( c = 3 \).
• Substitute these into the formula: \( x = \frac{-(-3) \pm \sqrt{-3}}{2 \cdot 1} = \frac{3 \pm \sqrt{-3}}{2} \).

Because the discriminant (\( -3 \)) is negative, we use complex numbers:

• \( \sqrt{-3} \) is transformed to \( i\sqrt{3} \), where \( i \) is the imaginary unit.
• Simplifying gives solutions: \( x_1 = \frac{3}{2} + \frac{i \sqrt{3}}{2} \) and \( x_2 = \frac{3}{2} - \frac{i \sqrt{3}}{2} \).

The quadratic formula thus yields the two complex solutions in the form of \( a + bi \).