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A quadratic equation is a second-degree polynomial equation in the form: ax^2 + bx + c = 0 where 'a,' 'b,' and 'c' are constants, and 'x' represents an unknown variable.
Quadratic equations are prevalent in many areas of mathematics and science, making it essential to be comfortable solving them.

One way to solve these equations is through factoring. But before reaching the factoring step, the equation must be in 'standard form.'

For example, consider the equation given in the problem: \[ x^2 - 10x = -24 \].

Our first task is to rewrite it so that one side equals zero:
Step 1: Add 24 to both sides:
\[ x^2 - 10x + 24 = 0 \].

Now, our quadratic equation is in standard form, ready for factoring.

Factoring involves breaking down a complex expression into simpler expressions (factors) that can be multiplied to retrieve the original expression again.

For quadratic equations, we look for factors of the form: \[ (x - p)(x - q) = 0 \].

Here, 'p' and 'q' are numbers that need to satisfy two conditions: 1. Their product (p*q) must equal the constant term 'c' (in this case, 24).2. Their sum (p+q) must equal the coefficient of the middle term 'b' (in this case, -10).

Let's find these two numbers:
-The possible pairs of factors for 24 are: (1, 24), (2, 12), (3, 8), (4, 6).
For our equation, the pair that sums up to -10 is (-4) and (-6):
\[ (-4) + (-6) = -10 \] and \[ (-4) * (-6) = 24 \].

So, the factored form is: \[ (x - 4)(x - 6) = 0 \].

Once we have the equation factored, the zero-factor property comes into play.

The zero-factor property states that if the product of two factors is zero, at least one of the factors must be zero.

Apply this property to our factors: \[ (x - 4)(x - 6) = 0 \].

This gives us two simpler equations to solve:
\[ x - 4 = 0 \] //…which simplifies to…// \[ x = 4 \]

And
\[ x - 6 = 0 \] //…which simplifies to…// \[ x = 6 \].

These are the solutions to the original quadratic equation. They make the equation true when substituted back into the original form.