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The quadratic formula is a fundamental tool used for solving quadratic equations of the form \(ax^2 + bx + c = 0\). Quadratic equations are second-degree polynomials, meaning the highest power of the variable \(x\) is 2. The quadratic formula provides a reliable method for finding the roots or solutions of any quadratic equation, no matter how complex it might seem. The formula is given by:

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

This handy equation allows you to find the values of \(x\) where the quadratic equation equals zero. Here, \(b^2 - 4ac\) is an important part of the formula known as the discriminant. After identifying the coefficients \(a\), \(b\), and \(c\), you can plug these values into the quadratic formula to get the solutions. It’s important to remember that the '±' sign indicates there can be two possible solutions for \(x\), which can be solved by considering both \(+\sqrt{b^2 - 4ac}\) and \(-\sqrt{b^2 - 4ac}\). This is why quadratics typically have two solutions.

The discriminant is a key part of the quadratic formula and is represented by \(b^2 - 4ac\). It helps in determining the nature and the number of solutions you can expect from a quadratic equation. Depending on the value of the discriminant, you can quickly gauge how many solutions there are and what type they will be:

• If the discriminant is greater than zero (\(b^2 - 4ac > 0\)), the quadratic equation has two distinct real roots.
• If the discriminant is equal to zero (\(b^2 - 4ac = 0\)), there is exactly one real root, meaning the parabola touches the x-axis at one point.
• When the discriminant is less than zero (\(b^2 - 4ac < 0\)), no real roots exist and the solutions are complex or imaginary numbers, meaning the equation never actually crosses the x-axis.

In our specific case, the discriminant was calculated as 29, which is greater than zero, indicating two distinct real solutions for the equation \(x^2 - x - 7 = 0\).

Coefficients are integral parts of quadratic equations and play a crucial role in using the quadratic formula. In a quadratic equation of the form \(ax^2 + bx + c = 0\):

• \(a\) is the coefficient of \(x^2\) and is known as the quadratic coefficient. It determines the opening of the parabola (upward or downward).
• \(b\) is the coefficient of \(x\), called the linear coefficient, affecting the vertex and axis of symmetry of the parabola.
• \(c\) is the constant term, impacting the vertical position or the y-intercept of the parabola.

For our equation \(x^2 - x - 7 = 0\), we identified \(a = 1\), \(b = -1\), and \(c = -7\). Recognizing these values is important because they are used directly in calculating both the discriminant and the solutions through the quadratic formula. Understanding how each coefficient influences the graph of a quadratic equation can provide deeper insights into the behavior and solutions of the equation.