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A quadratic equation is any polynomial equation of degree two. It usually has the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). These equations are important in mathematics for determining the values, or roots, of \( x \) that satisfy the equation.
When solving a quadratic equation, the coefficients \( a \), \( b \), and \( c \) guide us in analyzing and solving it efficiently. For instance, in the equation \( x^2 - 6x + 10 = 0 \), \( a = 1 \), \( b = -6 \), and \( c = 10 \).
Quadratic equations can have two distinct roots, a single double root, or a pair of complex roots, depending on the value of another important part called the discriminant.

The discriminant is a specific value that helps determine the nature of the roots of a quadratic equation. It is derived from the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \) and is given by:\
\[ D = b^2 - 4ac \]
The discriminant tells us a lot:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), it has one real root, or a repeated real solution.
• If \( D < 0 \), the roots are complex and come as a conjugate pair.

In our equation \( x^2 - 6x + 10 = 0 \), we calculate the discriminant as \( D = (-6)^2 - 4 \times 1 \times 10 = 36 - 40 = -4 \), indicating the roots will be complex numbers.

The quadratic formula is a critical tool for finding the roots of any quadratic equation. Once we know the discriminant, the quadratic formula can help us directly calculate the roots. It is expressed as:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
This formula allows us to plug in the values for \( a \), \( b \), and \( D \), finding the specific values of \( x \) that solve the equation. For the equation \( x^2 - 6x + 10 = 0 \), with \( a = 1 \), \( b = -6 \), and \( D = -4 \), the roots are derived as:
\[ x = \frac{6 \pm \sqrt{-4}}{2} \]
To simplify the square root of a negative number, we integrate complex numbers.

Complex roots occur when the discriminant \( D \) of a quadratic equation is negative. These roots are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \).
For the equation \( x^2 - 6x + 10 = 0 \), the discriminant is -4, so the roots are complex. The square root of \( -4 \) is \( 2i \), as \( i \) allows us to write \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \).
Using the quadratic formula, we got the solutions:

• \( x_1 = \frac{6 + 2i}{2} = 3 + i \)
• \( x_2 = \frac{6 - 2i}{2} = 3 - i \)

These solutions are in the standard complex number form \( a + bi \), representing the conjugate pair of complex roots for the given equation.