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Grasping the methodology of solving quadratic equations is key to conquering many algebraic challenges. Typically, a quadratic equation looks like this: \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. The solution to such an equation unravels the values of \( x \) for which the equation holds true. To solve quadratic equations, three main methods are predominantly used:

• Factoring the quadratic equation when possible.
• Using the quadratic formula, which reliably works for all types of quadratic equations.
• Completing the square, which is a step-by-step process to manipulate the equation into a perfect square trinomial.

In the provided exercise, we approach the problem by first rearranging it into the standard form and eventually applying the quadratic formula for a definitive solution.

The quadratic formula is a powerful tool in algebra that provides the solutions to any quadratic equation. The formula is \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. It leverages the coefficients of the standard form of the quadratic equation, which are \( a \), \( b \), and \( c \). Here’s a breakdown of the steps using the formula:

• Identify and substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula.
• Calculate the discriminant, \( b^2 - 4ac \), to determine the nature of the roots.
• Apply arithmetic operations to solve for \( x \).

This formula is incredibly efficient as it consistently yields the roots regardless of whether the quadratic equation can be factored or not.

Understanding the standard form of a quadratic equation is vital because it is the starting point for various solution methods. The standard form is given as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are known values with \( a \) not equal to zero. When an equation is presented in a different structure, such as \( y(y+8)=-15 \), the first step involves manipulating it to resemble the standard form. This is essential because it allows for clarity in identifying the coefficients needed for methods such as factoring, completing the square, or using the quadratic formula. In the exercise, we expanded and rearranged the given equation to fit this standard form, which subsequently enabled us to identify the coefficients and proceed with the quadratic formula for a solution.