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Factoring is a fundamental method used to solve quadratic equations. The goal is to express a quadratic equation in the form * ax^2 + bx + c = 0 *in a more manageable expression, usually by breaking it down into two binomial expressions. In the given exercise, we have:\[ x^2 - 10x + 24 = 0 \]The challenge is to identify two numbers whose product equals the constant term (24 in this case) and whose sum equals the linear coefficient (-10 here). These numbers are -4 and -6. Thus, the factored form of the equation becomes:\[ (x - 4)(x - 6) = 0 \]Factoring helps to simplify the equation, making it easier to solve by finding the roots in the next steps.

Finding the roots of a quadratic equation involves determining the values of x that satisfy the equation. Once a quadratic equation is factored, finding its roots becomes straightforward. Consider the factored expression of our equation:\[ (x - 4)(x - 6) = 0 \]To find the roots, we apply the zero product property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero. This provides us two separate equations:- * x - 4 = 0 *- * x - 6 = 0 *Solving these gives us the roots:- When * x - 4 = 0 *, then * x = 4 *- When * x - 6 = 0 *, then * x = 6 *These values of x are the solutions to the original quadratic equation.

Solving a quadratic equation means finding all possible values of the variable that make the equation true. For quadratic equations, this typically involves reducing the equation to a simpler form, such as factoring, and then using techniques like the zero product property.

In our example, after factoring the equation \( (x - 4)(x - 6) = 0 \), we identified the solutions by setting each factor equal to zero. This method is particularly effective when the quadratic can be easily factored.

Overall, solving quadratic equations can also involve other methods like:

• Completing the square
• Using the quadratic formula
• Graphical approaches

Choosing the right method often depends on the specific form of the quadratic equation you are dealing with. However, in our case, factoring provides a quick path to the solution.