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Consider $8x^2 - 48x = -104$.


  1. Write the equation so that $a = 1$:

$x^2 + [\text{ _ }]x = [\text{ _ }]$


  1. Complete the square:

$x^2 - 6x + [\text{ _ }] = -13 + [\text{ _ }]$


  1. Factor the trinomial and simplify:

$(x + [\text{ _ }])^2 = [\text{ _ }]$

Verified Answer :

  • Divide the initial equation \(8x^2 - 48x = -104\) by 8 to obtain \(x^2 - 6x = -13\).
  • Determine the value needed to complete the square by calculating the square of half the linear coefficient, which is \((\frac{-6}{2})^2 = 9\).
  • Add 9 to both sides of the equation: \(x^2 - 6x + 9 = -13 + 9\).
  • Factor the left side and simplify the right side to reach the final form \((x - 3)^2 = -4\).
  • The final equation is \(\boxed{(x - 3)^2 = -4}\).

Explanation

  1. Problem Analysis and Objective

The objective is to rewrite the quadratic equation \(8x^2 - 48x = -104\) into the completed square form \((x + \square)^2 = \square\). We begin by ensuring the leading coefficient of the quadratic term \(x^2\) is 1. We then isolate the quadratic and linear terms and add a specific constant to both sides of the equation to create a perfect square trinomial. Finally, we factor the trinomial and simplify the constant on the right side to reach the desired form.

  1. Rewriting the Equation to Monic Form

First, divide every term in the given equation \(8x^2 - 48x = -104\) by 8 to make the coefficient of \(x^2\) equal to 1:
\[\frac{8x^2}{8} - \frac{48x}{8} = \frac{-104}{8}\]
\[x^2 - 6x = -13\]

  1. Completing the Square

To complete the square for the quadratic expression \(x^2 - 6x\), we need to add the square of half the coefficient of \(x\). The coefficient of \(x\) is \(-6\). Half of \(-6\) is \(-3\), and its square is \((-3)^2 = 9\). Adding 9 to both sides of the equation gives:
\[x^2 - 6x + 9 = -13 + 9\]

  1. Factoring and Simplifying

The expression on the left side, \(x^2 - 6x + 9\), is a perfect square trinomial that factors into \((x - 3)^2\). On the right side, we perform the arithmetic: \(-13 + 9 = -4\). Thus, the equation in the required form is:
\[(x - 3)^2 = -4\]

Examples

Completing the square is a fundamental algebraic technique used in physics and engineering to analyze projectile motion. For instance, when calculating the maximum height of a launched object, the path of the projectile is described by a downward-opening parabola. By rewriting the standard quadratic equation into vertex form using the completing the square method, we can easily identify the vertex, which corresponds to the maximum or minimum point in real-world scenarios. This technique simplifies the analysis of physical systems, allowing for precise predictions in everything from sports ball trajectories to structural engineering designs.

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