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The factored form of a quadratic equation is a way of expressing the equation as a product of its linear factors. This is written as \[ (x - p)(x - q) = 0 \] where \( p \) and \( q \) are the roots of the equation.
To find the factored form, you need the roots of the equation. These roots are simply the values of \( x \) that make the equation equal to zero.
In our example, the roots are given as -3 and 5, so the factored form becomes \[(x + 3)(x - 5) = 0\].
The factored form gives you a direct insight into the roots and is very useful for solving quadratic equations as it breaks down the equation into simpler parts.

The roots of a quadratic equation, sometimes called solutions or zeros, are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \).
These roots can be real or complex numbers.
For quadratic equations, the roots can be found using a variety of methods such as factoring, completing the square, or utilizing the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
In our particular problem, the solutions were stated directly as -3 and 5.
Knowing the roots allows us to easily write the factored form of the equation, which can then be expanded or manipulated to suit different purposes.

The general form of a quadratic equation is represented as \[ ax^2 + bx + c = 0 \].
It is the standard way to express quadratics and is useful for a range of different mathematical applications including graphing and solving.
Starting from the factored form, such as \[ (x + 3)(x - 5) = 0\], we can expand it to move to the general form.
This involves multiplying the factors together, which gives us \[ x^2 - 5x + 3x - 15 = 0 \], then combining like terms leads to \[ x^2 - 2x - 15 = 0 \].
This is now the general form of the quadratic equation with coefficients \( a = 1 \), \( b = -2 \), and \( c = -15 \). Understanding how to transition between different forms of a quadratic equation can greatly enhance problem-solving capabilities.