91Ó°ÊÓ

The discriminant is a key concept in understanding quadratic equations. It helps us determine the nature of the roots without actually solving the entire equation. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by the expression \( b^2 - 4ac \).

This value is crucial because it tells us whether the solutions to the quadratic equation are real or complex:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, as in the above exercise where \( b^2 - 4ac = -32 \), there are no real roots. Instead, the equation has two complex roots, which involve imaginary numbers.

Understanding the discriminant allows us to anticipate the type of solutions we can expect, providing a quick way to assess the nature of the roots.

When the discriminant is negative, as it is in the exercise with a value of \(-32\), the quadratic equation does not intersect the x-axis, which means it has no real roots. Instead, it has complex roots. Complex roots arise from the square root of a negative number which involves the imaginary unit \( i \), where \( i^2 = -1 \).

Using the quadratic formula, the roots can be expressed as \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In the case of a negative discriminant, you'll have \( \pm \sqrt{-32} \), which converts to \( 4\sqrt{2}i \) (since \( \sqrt{-32} = 4\sqrt{2}i \)). Therefore, the solutions include both real and imaginary parts:

• Real part: \( \frac{1}{3} \)
• Imaginary part: \( \frac{2\sqrt{2}}{3}i \)

Thus, the two complex roots are \( m = \frac{1}{3} + \frac{2\sqrt{2}}{3}i \) and \( m = \frac{1}{3} - \frac{2\sqrt{2}}{3}i \). By grasping this concept, one can solve and understand how quadratic equations behave beyond real number solutions.

The quadratic formula is a powerful tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides a direct way to find the roots of the equation, which are the values of \( x \) that satisfy the equation.

The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how it works:

• The term \(-b\) indicates the factor opposite to the linear coefficient.
• The \( \pm \) symbol highlights the possibility of two solutions: one with addition and one with subtraction.
• The expression \( \sqrt{b^2 - 4ac} \) is the discriminant discussed earlier, central to determining the root nature.
• Everything is divided by \( 2a \), which is twice the coefficient of \( x^2 \).

When the discriminant is negative, the formula naturally accommodates complex numbers, as evidenced by solutions involving the imaginary unit \( i \). The quadratic formula, therefore, not only helps solve for real roots but also elegantly handles complex roots, providing a comprehensive solution approach for quadratic equations.