91Ó°ÊÓ

A quadratic equation is generally written in the form \( ax^2 + bx + c = 0 \). Finding real solutions means identifying \( x \)-values that satisfy the equation using real numbers. These solutions are the points where the graph of the equation intersects the x-axis.

• If a quadratic equation has two distinct real solutions, its graph crosses the x-axis at two places.
• If it has one real solution, the graph touches the x-axis at a single point, also called a "double root" or "repeated root".
• If there are no real solutions, the graph doesn't touch or cross the x-axis.

In this context, knowing whether real solutions exist or not is crucial, and that's where the discriminant plays a critical role in our analysis of the solutions.

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is denoted by the symbol \( \Delta \), calculated as \( b^2 - 4ac \). It provides valuable insights into the nature of the solutions of the equation without actually solving it.

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution (a double root).
• If \( \Delta < 0 \), the equation has no real solutions, but instead, it has complex solutions with real and imaginary parts.

In our exercise, the discriminant was calculated as \(-51\). Since this value is less than zero, it indicates that the quadratic equation \( 5x^2 - 7x + 5 = 0 \) does not have any real solutions but instead has complex solutions.

Complex numbers extend our understanding of numbers beyond the real number system. They are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined as \( i^2 = -1 \).
Complex solutions of quadratic equations arise when the discriminant \( \Delta \) is negative. In such cases, solutions can be written as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Since \( \sqrt{b^2 - 4ac} \) will yield an imaginary number when \( b^2 - 4ac < 0 \), the solutions will include the imaginary unit \( i \).

• These complex solutions appear as conjugate pairs, \( a + bi \) and \( a - bi \).
• Understanding complex numbers helps in solving problems where real numbers are not sufficient.

For our specific quadratic equation, the lack of real solutions means we need to calculate and express them using complex numbers, enhancing our ability to solve a broader spectrum of equations.