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$Rewrite the equation by completing the square. {:[4x^(2)-4x+1=0],[(x+◻)^(2)=◻]:}$

Rewrite the equation by completing the square. $\newline$ $4x^{2}-4x+1=0$ $\newline$ $(x+\square)^{2}=\square$

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Math Problems

Algebra 1

Solve a quadratic equation by completing the square

Full solution

Q. Rewrite the equation by completing the square.

\newline

4x^{2}-4x+1=0

\newline

(x+\square)^{2}=\square

Given equation: Start with the given equation. $4x^2 - 4x + 1 = 0$
Factor out coefficient: Factor out the coefficient of $x^2$ from the first two terms on the left side. $\newline$ $4(x^2 - x) + 1 = 0$
Complete the square: To complete the square, find the value that needs to be added and subtracted to the expression inside the parentheses to make it a perfect square trinomial. This value is $(\frac{b}{2a})^2$ , where $a$ is the coefficient of $x^2$ and $b$ is the coefficient of $x$ . $\newline$ In this case, $a = 1$ and $b = -1$ (after factoring out the $4$ ), so $(\frac{b}{2a})^2 = (\frac{-1}{2 \cdot 1})^2 = (\frac{1}{2})^2 = \frac{1}{4}$ .
Add and subtract: Add and subtract $(\frac{1}{4})$ inside the parentheses. Since we factored out a $4$ , we need to add and subtract $4\times(\frac{1}{4})$ to keep the equation balanced. $\newline$ $4(x^2 - x + \frac{1}{4} - \frac{1}{4}) + 1 = 0$ $\newline$ $4(x^2 - x + \frac{1}{4}) - 4\times(\frac{1}{4}) + 1 = 0$
Simplify the equation: Simplify the equation by combining like terms. $\newline$ $4(x^2 - x + \frac{1}{4}) - 1 + 1 = 0$ $\newline$ $4(x^2 - x + \frac{1}{4}) = 0$
Perfect square trinomial: The expression inside the parentheses is now a perfect square trinomial, which can be written as $(x - \frac{1}{2})^2$ . $\newline$ $4(x - \frac{1}{2})^2 = 0$
Isolate the perfect square: Divide both sides by $4$ to isolate the perfect square. $\newline$ $(x - \frac{1}{2})^2 = \frac{0}{4}$ $\newline$ $(x - \frac{1}{2})^2 = 0$