If \(x^2+12 x=64\) and \(x>0\), what is the value of \(x\) ? A) 2 B) 4 C) 8 D) 16

When faced with a quadratic equation such as \(x^2 + 12x = 64\), the first step is to transform it into the standard form \(ax^2 + bx + c = 0\). By moving all terms to one side, we obtain \(x^2 + 12x - 64 = 0\), setting the stage for the solution. Solving quadratic equations can often be done by factoring, completing the square, using the quadratic formula, or graphical methods.

Since factoring can be complex and tedious, particularly for large coefficients or when solutions aren't integers, the quadratic formula provides a straightforward alternative. Using the quadratic formula ensures that you'll find both real and complex solutions to the equation, if they exist. Therefore, it's a valuable tool for students who are preparing for standardized tests like the SAT, which may include problems requiring this method.

The quadratic formula is a master key for unlocking any quadratic equation. It is written as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this formula, \(a\), \(b\), and \(c\) represent the coefficients from the equation \(ax^2 + bx + c = 0\).

The symbol \(\pm\) indicates that there are usually two solutions—an outcome when plus is chosen (the 'plus' solution) and another for when minus is chosen (the 'minus' solution). Remember, the discriminant, \(b^2 - 4ac\), helps determine the nature of the roots. When it's positive, there are two distinct real solutions; if it's zero, there's one real solution; and if it's negative, there are two complex solutions.

This triumphant formula equips students with a consistent method for navigating these polynomial waters, being especially useful when other techniques like factoring fail or are hard to implement.

Quadratic equations are a common sight on the SAT, making their understanding key for test takers aiming for a high score. While quadratic equations on the SAT range from simple to complex, practicing with problems of varying difficulty helps build the confidence and proficiency needed on test day.

Students should focus on honing their skills in identifying quadratic equations, moving terms to one side to form the standard quadratic equation, selecting the appropriate method for solution, and then efficiently executing that method. Time management is crucial in the SAT, so knowing when to apply the quadratic formula over other methods can save precious time. Remember, verifying whether the solutions are positive, negative, or zeroes plays a significant role, as many SAT questions incorporate additional conditions, just as in the exercise provided.

In preparation for the SAT, students are encouraged to practice a plethora of problems, ensuring they are comfortable with all potential problem types and solution methods. This includes recognizing when to disregard extraneous or non-sensical solutions in the context of a problem, as seen when we dismiss the negative solution in our example due to specified conditions.