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The quadratic formula is a fundamental tool for solving quadratic equations. A quadratic equation is any equation that can take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula helps us find the solutions, also known as roots, of this equation. The formula itself is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The variables \( a \), \( b \), and \( c \) represent the coefficients in the equation. To utilize this formula effectively, we first identify these coefficients, substitute them into the formula, and simplify. The \( \pm \) sign indicates that there can be two solutions. These arise because taking the square root of a number can result in both a positive and a negative value. Applying the quadratic formula is a reliable method, especially when factoring the quadratic equation seems complex or is impossible.

The roots of a quadratic equation are the solutions that satisfy the equation \( ax^2 + bx + c = 0 \). These roots are the values for \( x \) that make the entire expression equal zero. Depending on the discriminant \( b^2 - 4ac \), which is the value under the square root in the quadratic formula, the nature of the roots can vary:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, also called a repeated or double root. This occurs when the parabola touches the x-axis at one point.
• If \( b^2 - 4ac < 0 \), the equation has two complex roots, which are not real numbers.

In our specific example, the discriminant is \( b^2 - 4ac = 49 - 48 = 1 \), which is greater than zero, indicating two distinct real roots. By evaluating the formula, we find that the roots are \(-3\) and \(-4\). Understanding the roots helps us describe the x-intercepts of the parabola represented by the quadratic equation.

Solving quadratic equations involves finding the values of the variable that make the equation true. There are several methods available for solving quadratics, each with its own applications and advantages:

• Factoring: An efficient method when the quadratic can be expressed as a product of two binomials, allowing for quick identification of the roots.
• Completing the Square: This technique rewrites the quadratic in a perfect square form, useful when factoring is complicated. It also underlies the derivation of the quadratic formula itself.
• Quadratic Formula: A universal tool applicable to all quadratic equations, regardless of their specific form or difficulty in factoring. It is especially useful when the roots are not integers.
• Graphing: Provides a visual representation of the solutions and is useful when approximating roots. The solutions correspond to the x-intercepts of the graph.

In our exercise, we applied the quadratic formula because it provides a straightforward, reliable solution. By substituting the coefficients into the formula, we determined that the roots are \(-3\) and \(-4\), offering a precise answer to the equation.