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The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a systematic way to find the values of \( x \) that satisfy the equation. The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula can solve any quadratic equation, regardless of whether or not it can be easily factored. Here's how it works:

• The term \(-b\) refers to the opposite of the coefficient \(b\) in the equation.
• The expression \(\sqrt{b^2 - 4ac}\) is called the discriminant, which we will discuss more in another section.
• The equation is divided by \(2a\), which allows you to accurately calculate the roots.

When using the quadratic formula, always remember to calculate the discriminant first to determine the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant signifies exactly one real root, and a negative discriminant suggests two complex roots. This property makes the quadratic formula particularly versatile.

The discriminant is a crucial component of the quadratic formula, represented by the expression \(b^2 - 4ac\). It helps to determine the nature and number of the roots of a quadratic equation. Let's break down its significance:

• A positive discriminant (\(b^2 - 4ac > 0\)) means the equation has two distinct real roots. This signifies that the parabola represented by the quadratic equation intersects the x-axis at two points.
• A discriminant equal to zero (\(b^2 - 4ac = 0\)) indicates one real double root. In this scenario, the parabola touches the x-axis at precisely one point, which is also known as a repeated or double root.
• A negative discriminant (\(b^2 - 4ac < 0\)) suggests the presence of two complex roots. This means the parabola does not intersect the x-axis at all. Instead, it lies entirely above or below the axis.

By evaluating the discriminant before applying the rest of the quadratic formula, you gain insight into what kind of solutions to expect, which makes solving quadratic equations more intuitive.

The roots of a quadratic equation are the solutions that satisfy the equation, meaning they are the values of \( x \) for which the equation is zero. Depending on the result from the discriminant, the nature of these roots varies:

• If the discriminant is positive, you have two distinct real roots. These are different values of \( x \) obtained by solving the quadratic formula with \(+\) and \(-\) in the expression \(\pm \sqrt{b^2 - 4ac}\).
• If the discriminant is zero, the equation has exactly one real root, or more formally, a repeated root. This occurs because \(\sqrt{b^2 - 4ac}\) becomes zero, simplifying the quadratic formula significantly.
• Negative discriminants lead to complex roots, often expressed in the form \( a \pm bi \), where \(i\) is the imaginary unit. This happens because the square root of a negative number introduces an imaginary component.

In our original problem, the equation \(x^2 - 13x + 36 = 0\) had a positive discriminant of 25, leading to two distinct real roots, which we found to be 9 and 4. These roots verify the equation because substituting them back in results in zero.