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Rewrite the equation by completing the square. {:[4x^(2)+20 x+25=0],[(x+◻)^(2)=◻]:}

Rewrite the equation by completing the square.\newline4x2+20x+25=04x^{2}+20x+25=0\newline(x+□)2=□(x+\square)^{2}=\square

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Full solution

Q. Rewrite the equation by completing the square.\newline4x2+20x+25=04x^{2}+20x+25=0\newline(x+□)2=□(x+\square)^{2}=\square
  1. Factor out coefficient of x^22: We are given the quadratic equation 4x2+20x+25=04x^2 + 20x + 25 = 0. To complete the square, we want to express the equation in the form (x+□)2=□(x + \square)^2 = \square. First, we need to factor out the coefficient of x2x^2 from the first two terms on the left side of the equation.\newline4(x2+5x)+25=04(x^2 + 5x) + 25 = 0
  2. Find number to fill the square: Next, we need to find a number to fill the square (□\square) that makes x2+5xx^2 + 5x into a perfect square trinomial. We take half of the coefficient of xx, which is 52\frac{5}{2}, and square it to get (52)2=254\left(\frac{5}{2}\right)^2 = \frac{25}{4}.\newline4(x2+5x+254)+25−4×254=04\left(x^2 + 5x + \frac{25}{4}\right) + 25 - 4 \times \frac{25}{4} = 0
  3. Add and subtract to balance equation: We add and subtract 254\frac{25}{4} inside the parentheses to keep the equation balanced. We then multiply the subtracted 254\frac{25}{4} by 44 to subtract it from the outside of the parentheses.\newline4(x2+5x+254)−25=04(x^2 + 5x + \frac{25}{4}) - 25 = 0
  4. Write as a squared binomial: Now we have a perfect square trinomial inside the parentheses, and we can write it as a squared binomial. 4(x+52)2−25=04(x + \frac{5}{2})^2 - 25 = 0
  5. Isolate the squared binomial: Finally, we add 2525 to both sides to isolate the squared binomial.\newline4(x+52)2=254(x + \frac{5}{2})^2 = 25
  6. Divide both sides to solve: To complete the square, we divide both sides by 44 to solve for the squared binomial.\newline(x+52)2=254(x + \frac{5}{2})^2 = \frac{25}{4}

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