Consider $8x^2 - 48x = -104$.
Write the equation so that $a = 1$: $x^2 + \quad x =$
The problem provides a quadratic equation \(8x^2 - 48x = -104\) and asks us to rewrite it in the standard form \(x^2 + bx = c\), where the coefficient of the quadratic term \(x^2\) is \(1\). We observe that the current equation has \(8\) as the coefficient of \(x^2\). Dividing the entire equation by this coefficient will simplify it into the desired form.
To transform the equation \(8x^2 - 48x = -104\) into the form \(x^2 + bx = c\), we divide both sides by \(8\):
\[\frac{8x^2 - 48x}{8} = \frac{-104}{8}\]
Distributing the division on the left side, we get:
\[\frac{8x^2}{8} - \frac{48x}{8} = \frac{-104}{8}\]
Performing the arithmetic:
\[x^2 - 6x = -13\]
This matches the requested form \(x^2 + bx = c\), where \(b = -6\) and \(c = -13\).
Having performed the division, we have successfully rewritten the equation as \(x^2 - 6x = -13\). Thus, the missing values in the expression \(x^2 + \boxed{\phantom{-6}}x = \boxed{\phantom{-13}}\) are \(-6\) and \(-13\), respectively. The final form is \(x^2 + \boxed{-6}x = \boxed{-13}\).
In algebra, manipulating equations is a fundamental skill used to isolate variables or simplify complex expressions. For instance, when solving for the dimensions of a rectangular garden given its area and a relationship between its length and width, you often need to rearrange a quadratic equation into a standard form to apply the quadratic formula or complete the square. By normalizing the coefficient of \(x^2\) to 1, you make it much easier to identify the coefficients needed for further algebraic techniques, ensuring a clean and manageable path to the solution.