Answered

Solve the quadratic equation by completing the square.

$x^2-2x=4$



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$x^2-2x=4 \Rightarrow \frac{1}{2}x^2-x+\square=2+\square$
B$x^2-2x=4 \Rightarrow x^2-2x+\square=4$
C$x^2-2x=4 \Rightarrow \frac{1}{2}x^2-x+\square=2$
D$x^2-2x=4 \Rightarrow x^2-2x+1=4+1$



The solution set is { $\square$ }.

(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)

Verified Answer :

  • Identify the coefficient of \(x\) as \(-2\), then calculate \(\left(\frac{-2}{2}\right)^2 = 1\) to complete the square.
  • Add \(1\) to both sides of \(x^2 - 2x = 4\) to get \(x^2 - 2x + 1 = 5\), which is option D.
  • Rewrite the equation as \((x - 1)^2 = 5\) and solve for \(x\) to get \(x = 1 \pm \sqrt{5}\).
  • The solution set is \(\boxed{\{1-\sqrt{5}, 1+\sqrt{5}\}}\).

Explanation

  1. Analyzing the Equation and Completing the Square

We are given the quadratic equation \(x^2 - 2x = 4\). Our goal is to solve this equation by completing the square. Completing the square involves transforming the quadratic expression \(x^2 + bx\) into a perfect square trinomial \((x + \frac{b}{2})^2\) by adding a constant \(\left(\frac{b}{2}\right)^2\) to both sides of the equation. Here, the coefficient of \(x\) is \(b = -2\). Therefore, we calculate \(\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1\). Adding 1 to both sides of the equation gives:
\[x^2 - 2x + 1 = 4 + 1\]
This corresponds to option D, which is the appropriate process for rearranging the equation to complete the square.

  1. Solving for x

With the equation in the form \(x^2 - 2x + 1 = 5\), we can now rewrite the left side as a perfect square:
\[(x - 1)^2 = 5\]
To solve for \(x\), we take the square root of both sides, keeping in mind that the square root of 5 can be positive or negative:
\[x - 1 = \pm\sqrt{5}\]
Adding 1 to both sides gives the solutions:
\[x = 1 \pm \sqrt{5}\]
Thus, the solutions are \(1 - \sqrt{5}\) and \(1 + \sqrt{5}\).

  1. Conclusion

The correct process for rearranging the equation is D. The solution set for the quadratic equation \(x^2 - 2x = 4\) is \(\{1-\sqrt{5}, 1+\sqrt{5}\}\). We have thus found both exact values for \(x\).

Examples

Completing the square is a powerful algebraic technique used in various real-life scenarios, such as in physics when determining the trajectory of an object. For instance, if the height of a projectile is given by a quadratic function of time, \(h(t) = at^2 + bt + c\), one can complete the square to find the vertex of the parabola, which represents the maximum height reached by the object and the time it takes to get there. This helps engineers and scientists predict and optimize paths for everything from sports projectiles to satellite launches.

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