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The quadratic formula is a powerful tool for finding the roots of any quadratic equation. A quadratic equation is typically of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula specifically allows us to calculate the values of \(x\) that satisfy the equation. Here's what the formula looks like:

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

The formula may look intimidating at first, but it becomes straightforward once you break it down. The letters \(b\) and \(a\) simply represent the coefficients from the equation you're working with. The symbol \(\pm\) indicates that there will generally be two solutions. The expression under the square root is known as the discriminant, which tells us a lot about the nature of the roots.

The discriminant is the part of the quadratic formula under the square root: \(b^2 - 4ac\). The value of the discriminant provides crucial information about the nature of the roots of a quadratic equation.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the equation has two complex roots, which are not real numbers.

In our example with the equation \(x^2 + 4x - 12 = 0\), the discriminant is \(64\) (calculated as \(b^2 - 4ac = 16 + 48\)), which is positive. Therefore, the equation has two distinct real roots. Calculating the discriminant first helps us anticipate these solutions before actually solving the equation with the quadratic formula.

The roots of a quadratic equation are the values of \(x\) that make the equation true, essentially where the graph of the equation will cross the \(x\)-axis. Using the quadratic formula provides a systematic method to find these roots. In our specific problem, the roots of the equation \(x^2 + 4x - 12 = 0\) come from substituting the values into the formula:

• \(x = \frac{-4 \pm \sqrt{64}}{2}\)
• \(\sqrt{64} = 8\), giving us \(x = \frac{-4 + 8}{2} = 2\) and \(x = \frac{-4 - 8}{2} = -6\)

Thus, the roots of the quadratic equation are \(x = 2\) and \(x = -6\). These solutions indicate where the parabola described by the equation intersects the \(x\)-axis. Understanding how to apply the quadratic formula and interpret its results is central to mastering quadratic equations.