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The quadratic formula is a crucial tool in algebra for finding the roots of a quadratic equation. A quadratic equation generally looks like this: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \) to ensure the equation is indeed quadratic. The quadratic formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The formula helps solve any quadratic equation by providing the values of \( x \) that satisfy the equation. It works for equations with real or complex roots. Use the formula by substituting the coefficients of your equation into \( a \), \( b \), and \( c \) in the formula. Remember, the \( \pm \) sign indicates you should solve it twice: once with a plus and once with a minus to get both potential roots.
This is why most quadratic equations have two solutions.

The discriminant is a key part of the quadratic formula; it helps determine the nature of the roots of a quadratic equation. Represented by \( D \), it is calculated as:\[ D = b^2 - 4ac \]The value of the discriminant gives vital insights:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is exactly one real root (repeated).
• If \( D < 0 \), the roots are complex or imaginary.

In our exercise, \( D = 196 \) which is greater than zero, indicating that the equation \( x^2 + 10x - 24 = 0 \) has two distinct real solutions. Thus, the discriminant is fundamental in understanding what kind of solutions to expect before even computing them.

Solving quadratic equations can initially seem challenging, but breaks down easily using a straightforward process. Follow these steps to solve using the quadratic formula:1. **Identify the coefficients**: Extract \( a \), \( b \), and \( c \) from the standard form of the quadratic equation.2. **Compute the discriminant**: Find the discriminant using \( b^2 - 4ac \).3. **Evaluate the square root of the discriminant**: This step is crucial if the discriminant is non-negative because you'll need its square root.4. **Plug into quadratic formula**: With the discriminant and coefficients, substitute into \[ x = \frac{-b \pm \sqrt{D}}{2a} \].5. **Calculate solutions**: Perform arithmetic to solve for \( x \) in both the positive and negative scenarios provided by the \( \pm \) operator.Returning to our equation \( x^2 + 10x - 24 = 0 \), it took identifying coefficients, computing the discriminant, and applying these steps to determine that \( x = 2 \) and \( x = -12 \). Each step ensures our calculations are spot-on for finding the right solutions.