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The quadratic formula is a powerful tool used to find the solutions, or roots, of a quadratic equation. A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \),
where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable.

The quadratic formula is expressed as:

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

This formula allows you to calculate the values of \( x \) that make the equation true by simply plugging in the coefficients \( a \), \( b \), and \( c \).

The term \( \pm \sqrt{b^2 - 4ac} \) indicates that there can be two potential values of \( x \), corresponding to each of the plus or minus signs. These will be our potential roots, and understanding this helps solve quadratic inequalities and equations quickly.

The discriminant is a specific part of the quadratic formula that provides essential insights into the nature of the roots.
It is given by the expression \( b^2 - 4ac \).

We can use the discriminant to determine the nature of the solutions as follows:

• If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, implying a perfect square trinomial.
• If \( b^2 - 4ac < 0 \), there are no real roots, indicating two complex or imaginary roots.

In our exercise, the discriminant was calculated as \( 89 \), a positive number. This indicated that the quadratic equation \( -x^2 + 7x + 10 = 0 \) has two real roots, essential for solving the inequality at hand.

Interval notation is a method used to describe a set of numbers between two endpoints on a number line. It is highly efficient for expressing the solutions of inequalities.

In interval notation:

• Parentheses \( (a, b) \) mean that the endpoints \( a \) and \( b \) are not included in the interval.
• Brackets \( [a, b] \) mean that the endpoints are included.

Additionally,

• \(( -\infty, a )\) or \(( a, \infty )\) represents an interval that extends infinitely in one direction, with \(-\infty\) or \(\infty\) never included, hence always using parentheses.

For our inequality \( -x^2 + 7x + 10 \geq 0 \), the solution expressed in interval notation is \( [-1.21, 8.21] \), indicating that the values between and including \(-1.21\) and \(8.21\) satisfy the inequality.

The roots of a quadratic equation are the values of \( x \) that solve the equation \( ax^2 + bx + c = 0 \).
These roots are where the graph of the quadratic equation crosses the x-axis in a coordinate plane.

Finding these roots can be done using:

• The quadratic formula, as shown in the section above.
• Factoring the quadratic if it is easily factorable.
• Completing the square, which is a more complex algebraic technique.

For the equation \( -x^2 + 7x + 10 = 0 \), we used the quadratic formula to find our roots: \( x_1 \approx -1.21 \) and \( x_2 \approx 8.21 \).
These roots help us in creating intervals to test which parts of the domain satisfy the original inequality.