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The quadratic formula is a fundamental tool in algebra that allows us to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). It is expressed as:

• \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides a direct method to find the roots (solutions) of any quadratic equation by substituting the appropriate coefficients \( a \), \( b \), and \( c \).
In using this formula, there are usually three possible scenarios for the nature of the roots:

• If the discriminant (explained later) is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, as in the given example, the roots are complex numbers (involving the imaginary unit \( i \)).

By plugging the values of \( a = -10 \), \( b = 4 \), and \( c = -2 \) into the quadratic formula, we demonstrated that the equation has complex roots.

The discriminant is an important algebraic expression found within the quadratic formula under the square root symbol: \( b^2 - 4ac \).It provides critical information about the nature of the roots of the quadratic equation.

Understanding the Discriminant:

- If the discriminant is positive, the quadratic will have two distinct real roots.- If the discriminant is zero, there will be one real root, which means the parabola touches the x-axis at exactly one point.- If the discriminant is negative, the roots are not real but rather complex numbers. These roots occur in pairs as complex conjugates.
In the exercise, after computing the discriminant \( D \) as \( 4^2 - 4(-10)(-2) = -64 \),we confirmed that it is negative, indicating that our quadratic equation does not have real roots, but instead has complex conjugates.

Why it Matters:

The sign of the discriminant is crucial in predicting the type of solutions without solving the equation fully, saving time and effort in many problem-solving scenarios.

Complex roots appear when the discriminant of a quadratic equation is negative.These solutions are known as complex conjugates.Each complex number has a corresponding conjugate that mirror its real part but has an opposite imaginary part.

Definition:

A complex number is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part (with \( i \) being the square root of \(-1\)).A complex conjugate takes the form \( a - bi \).
In our quadratic equation example, the computed roots are \( \frac{1}{5} + \frac{2i}{5} \) and \( \frac{1}{5} - \frac{2i}{5} \).These solutions clearly show the complex conjugate relationship:

• The real parts \( \frac{1}{5} \) are the same.
• The imaginary parts are \( +\frac{2i}{5} \) and \( -\frac{2i}{5} \).

Complex conjugates are important because they maintain the symmetry and balance in solving polynomial equations, ensuring solutions remain within the complex number system.