Solve each equation in Exercises 47–64 by completing the square. $4 x^{2} - 4 x - 1 = 0$

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Start with the given quadratic equation: $4x^2 - 4x - 1 = 0$.

Divide the entire equation by 4 to make the coefficient of $x^2$ equal to 1: $x^2 - x - \frac{1}{4} = 0$.

Move the constant term to the right side: $x^2 - x = \frac{1}{4}$.

To complete the square, take half of the coefficient of $x$ (which is $-1$), square it, and add it to both sides. Half of $-1$ is $-\frac{1}{2}$, and its square is $\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$, so add $\frac{1}{4}$ to both sides: $x^2 - x + \frac{1}{4} = \frac{1}{4} + \frac{1}{4}$.

Rewrite the left side as a perfect square trinomial: $\left(x - \frac{1}{2}\right)^2 = \frac{1}{2}$, and then proceed to solve for $x$ by taking the square root of both sides.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves creating a binomial squared expression on one side, making it easier to solve for the variable by taking square roots.

Quadratic Equation Standard Form

A quadratic equation is typically written as ax² + bx + c = 0, where a, b, and c are constants. Understanding this form is essential because completing the square requires isolating the x terms and manipulating the equation accordingly.

Isolating the Variable

Before completing the square, it is important to isolate the quadratic and linear terms on one side of the equation and, if necessary, divide through by the coefficient of x². This step simplifies the process of forming a perfect square trinomial.

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