91Ó°ÊÓ

The quadratic formula is a powerful tool used to find the solutions to a quadratic equation, which takes the standard form of:\[ ax^2 + bx + c = 0 \]The solutions of a quadratic equation can be determined by the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, the expression \( b^2 - 4ac \) within the quadratic formula is known as the "discriminant." The discriminant helps us to understand the nature of the roots of the quadratic equation:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), the equation has exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the equation has two complex roots.

In the given exercise, we specifically need to evaluate the discriminant part of the quadratic formula.

Square roots are unique numbers that, when multiplied by themselves, produce the original number. The square root of a number \( n \) is represented as \( \sqrt{n} \). For example, \( \sqrt{25} = 5 \), because 5 multiplied by itself equals 25.
Evaluating square roots is a crucial step in both solving quadratic equations using the quadratic formula and determining the nature of the roots by examining the discriminant \( b^2 - 4ac \).
In our exercise, once we calculated the expression inside the square root, which is \( 25 \), we found that the square root is 5. This indicates that in this case, if part of a quadratic solution, the equation would have a real solution since the square root results in a real number.

Substitution is a fundamental technique in algebra where specific numbers are inserted in place of variables to simplify or solve equations. This approach allows for the accurate evaluation of expressions.
In the given exercise, we used substitution to input the values of \( a = 3 \), \( b = -1 \), and \( c = -2 \) into the expression \( b^2 - 4ac \). Here’s how it works:

• Identify the variables and their values, in this case: \( a = 3 \), \( b = -1 \), \( c = -2 \).
• Replace the variables in the expression with their respective values to get: \( (-1)^2 - 4(3)(-2) \).
• Compute the combined result to simplify the expression before moving on to further operations like square roots.

This substitution led us to the correct value of \( \sqrt{b^2 - 4ac} \), allowing us to solve the exercise accurately.