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$Rewrite the equation by completing the square. {:[2x^(2)-3x-5=0],[(x+◻)^(2)=◻]:}$

Rewrite the equation by completing the square. $\newline$ $2x^{2}-3x-5=0$ $\newline$ $(x+\square)^{2}=\square$

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Solve a quadratic equation by completing the square

Full solution

Q. Rewrite the equation by completing the square.

\newline

2x^{2}-3x-5=0

\newline

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

Given equation: Start with the given equation. $2x^2 - 3x - 5 = 0$
Divide by coefficient of x^ $2$ : Divide all terms by the coefficient of x^ $2$ to make the coefficient of x^ $2$ equal to $1$ . $\newline$ $x^2 - \left(\frac{3}{2}\right)x - \left(\frac{5}{2}\right) = 0$
Move constant term to right side: Move the constant term to the right side of the equation. $x^2 - \left(\frac{3}{2}\right)x = \left(\frac{5}{2}\right)$
Complete the square: To complete the square, add the square of half the coefficient of $x$ to both sides of the equation. The coefficient of $x$ is $-\frac{3}{2}$ , so half of that is $-\frac{3}{4}$ . The square of $-\frac{3}{4}$ is $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$ . $\newline$ $x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{5}{2} + \frac{9}{16}$
Combine terms on right side: Find a common denominator to combine the terms on the right side of the equation. $\newline$ $x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{40}{16} + \frac{9}{16}$ $\newline$ $x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{49}{16}$
Write left side as perfect square: Write the left side of the equation as a perfect square. $(x - \frac{3}{4})^2 = \frac{49}{16}$
Rewritten equation: The equation is now rewritten by completing the square. $\newline$ $(x + \square)^2 = \square$ becomes $(x - \frac{3}{4})^2 = \frac{49}{16}$

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