91Ó°ÊÓ

The standard quadratic form is an essential concept when it comes to solving quadratic equations. Any quadratic equation can be written in the standard form, which is expressed as
\[ ax^2 + bx + c = 0, \]
where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. This form is considered 'standard' because it provides a uniform structure that makes it easier to apply various methods of solving quadratics, such as factoring, completing the square, or using the quadratic formula.

Given the exercise \(3x^2-4x=15\), the first step to solving by factoring is to rewrite it into standard form. We do this by moving all terms to one side of the equation to get zero on the other side. Subtract 15 from both sides to find \(3x^2 - 4x - 15 = 0\). Now, the equation is ready to be factored as it represents the standard form.

Factoring is a way of breaking down a quadratic expression into simpler multiplicative components, also known as factors. To factor a quadratic expression, one must look for two binomials that, when multiplied together, give the original quadratic.

The task is to find two numbers that both add up to the coefficient \(b\) of the linear term and multiply to get the constant term \(c\) multiplied by the coefficient \(a\) of the quadratic term. For the example given, the target numbers should multiply to \(-3 \times 15 = -45\) and add up to \(-4\).

After identifying the numbers, in this case, \(-5\) and \(9\), you can then write the original quadratic expression as a product of two binomials: \[ (3x - 5)(x + 3) = 0. \]
This method simplifies the process of finding the solutions to the quadratic equation.

The zero product property is a critical principle used in algebra to solve equations. It states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if \[ ab = 0, \]
then either \(a = 0\) or \(b = 0\), or both. This property is immensely valuable when we have factored a quadratic equation because it allows us to break down the problem of finding the roots of the entire equation into simpler problems of finding the zeros of individual factors.

Applying this to our factored quadratic equation \[ (3x - 5)(x + 3) = 0, \]
it implies that \(3x - 5\) must be zero or \(x + 3\) must be zero. Solving for \(x\) in both scenarios gives us the equation's solutions: \[ x = \frac{5}{3} \quad \text{or} \quad x = -3. \]
Thus, using the zero product property, we have efficiently determined the values of \(x\) which satisfy the original quadratic equation.