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Rewrite the equation by completing the square. {:[x^(2)+x-72=0],[(x+◻)^(2)=◻]:}

Rewrite the equation by completing the square.\newlinex2+x−72=0(x+□)2=□ \begin{array}{l} x^{2}+x-72=0 \\ (x+\square)^{2}=\square \end{array}

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Full solution

Q. Rewrite the equation by completing the square.\newlinex2+x−72=0(x+□)2=□ \begin{array}{l} x^{2}+x-72=0 \\ (x+\square)^{2}=\square \end{array}
  1. Move constant term: To complete the square, we first need to move the constant term to the other side of the equation.\newlinex2+x−72=0x^2 + x - 72 = 0\newlineAdd 7272 to both sides to isolate the x terms.\newlinex2+x=72x^2 + x = 72
  2. Find number to complete the square: Next, we need to find a number to add to both sides of the equation to complete the square. This number is found by taking half of the coefficient of xx, squaring it, and adding it to both sides.\newlineThe coefficient of xx is 11, so half of it is 12\frac{1}{2}, and squaring it gives us (12)2=14(\frac{1}{2})^2 = \frac{1}{4}.\newlinex2+x+(14)=72+(14)x^2 + x + \left(\frac{1}{4}\right) = 72 + \left(\frac{1}{4}\right)
  3. Factor perfect square trinomial: Now we have a perfect square trinomial on the left side of the equation, which can be factored into (x+12)2(x + \frac{1}{2})^2.\newline(x+12)2=72+14(x + \frac{1}{2})^2 = 72 + \frac{1}{4}
  4. Simplify right side: To simplify the right side of the equation, we need to convert 7272 to a fraction with a denominator of 44 to combine it with 14\frac{1}{4}. Since 7272 is the same as 2884\frac{288}{4}, we can write:\newline(x+12)2=2884+14(x + \frac{1}{2})^2 = \frac{288}{4} + \frac{1}{4}
  5. Combine fractions: Combine the fractions on the right side of the equation.\newline(x+12)2=288+14(x + \frac{1}{2})^2 = \frac{288 + 1}{4}\newline(x+12)2=2894(x + \frac{1}{2})^2 = \frac{289}{4}

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