- Simplify the equation by dividing by \(8\), yielding \(x^2 - 6x = -13\).
- Identify \(A = -6\) and \(B = -13\).
- Complete the square by adding \((\frac{-6}{2})^2 = 9\) to both sides, resulting in \(x^2 - 6x + 9 = -13 + 9\).
- Final box values are \(A = -6, B = -13\), and the completion values are \(9\) and \(9\).
- Final equation format:
\(x^2 - 6x = -13\)
\(x^2 - 6x + \boxed{9} = -13 + \boxed{9}\)
Explanation
- Simplifying the Equation
The given equation is \(8x^2 - 48x = -104\). To isolate the quadratic term \(x^2\), we first divide the entire equation by the leading coefficient, which is \(8\).
Dividing \(8x^2\) by \(8\) gives \(x^2\).
Dividing \(-48x\) by \(8\) gives \(-6x\).
Dividing \(-104\) by \(8\) gives \(-13\).
Thus, the equation becomes \(x^2 - 6x = -13\). Comparing this to the form \(x^2 + Ax = B\), we find \(A = -6\) and \(B = -13\).
- Completing the Square
To complete the square for the expression \(x^2 - 6x\), we need to add a constant to both sides of the equation. We take the coefficient of \(x\), which is \(-6\), divide it by \(2\) to get \(-3\), and then square the result: \((-3)^2 = 9\). Adding \(9\) to both sides of the equation \(x^2 - 6x = -13\), we obtain \(x^2 - 6x + 9 = -13 + 9\). Comparing this to the target form, the missing values are both \(9\).
Examples
Completing the square is a fundamental algebraic technique used to solve quadratic equations and graph parabolas. For instance, if you are designing the trajectory of a ball or modeling the profit of a business based on production levels, you can transform complex quadratic functions into vertex form: \(f(x) = a(x - h)^2 + k\). This form makes it easy to identify the "vertex" or the maximum/minimum point of the graph, allowing you to quickly determine the optimal conditions for your model. Understanding this method is essential for advanced algebra, calculus, and physics.