Answered

Solve the quadratic equation by completing the square.

$x^2-6 x=1$



Choose the appropriate process for rearranging the equation, if needed, before completing the square to obtain an equation of the form $(x+k)^2=d$.

(Type an integer or a simplified fraction.)


A$x^2-6 x=1 \Rightarrow x^2-6 x+\square=1$
B$x^2-6 x=1 \Rightarrow \frac{1}{6} x^2-x+\square=\frac{1}{6}+\square$
C$x^2-6 x=1 \Rightarrow x^2-6 x+\square=1+\square$
D$x ^2-6 x =1 \Rightarrow \frac{1}{6} x ^2- x +\square=\frac{1}{6}$

Verified Answer :

  • Identify the coefficient of \(x\) as \(-6\) and determine the value to add: \(\left(\frac{-6}{2}\right)^2 = 9\).
  • Choose the correct strategy: adding \(9\) to both sides, which corresponds to option C: \(x^2 - 6x + 9 = 1 + 9\).
  • Simplify to \((x-3)^2 = 10\) and solve by taking the square root.
  • The solutions are \(\boxed{3 \pm \sqrt{10}}\).

Explanation

  1. Analyzing the Problem

We are tasked with solving the quadratic equation \(x^2 - 6x = 1\) by completing the square. The goal is to transform this equation into the form \((x+k)^2 = d\). The method of completing the square relies on identifying a constant term to add to both sides of the equation so that the left side becomes a perfect square trinomial. Given the equation \(x^2 - 6x = 1\), we identify the coefficient of the \(x\) term, which is \(-6\). The standard approach to create a perfect square trinomial \(x^2 + bx + c\) is to choose \(c = (b/2)^2\).

  1. Determining the Correct Rearrangement Strategy

In our equation \(x^2 - 6x = 1\), the coefficient of \(x\) is \(b = -6\). To complete the square, we calculate the term to add: \[\left(\frac{b}{2}\right)^2 = \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9.\] To maintain the equality of the equation, we must add this value, \(9\), to both sides. This leads to the transformation: \[x^2 - 6x + 9 = 1 + 9.\] Comparing this to the given choices, we see that it matches option C: \(x^2 - 6x + \square = 1 + \square\) where \(\square = 9\).

  1. Solving the Equation

Now that we have successfully rearranged the equation to \(x^2 - 6x + 9 = 1 + 9\), we can simplify the expression. The left side is a perfect square trinomial, \((x-3)^2\), and the right side is \(1 + 9 = 10\). So, the equation becomes \((x-3)^2 = 10\). To find the roots, we take the square root of both sides: \[x-3 = \pm\sqrt{10}.\] This yields two possible values for \(x\): \(x = 3 + \sqrt{10}\) and \(x = 3 - \sqrt{10}\). Thus, completing the square allows us to find the solutions \(\boxed{3 \pm \sqrt{10}}\).

Examples

Completing the square is a powerful technique for solving quadratic equations and is fundamental in understanding the geometry of parabolas. For instance, if you are designing a parabolic reflector for a telescope or a signal receiver, the vertex form of the parabola \(y = a(x-h)^2 + k\) is essential for locating the focus. By completing the square, you can easily shift from the general form \(y = ax^2 + bx + c\) to this vertex form, allowing you to accurately pinpoint where your signal will converge. This skill is foundational in fields like engineering, physics, and even architecture.

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