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Complete the square to re-write the quadratic function in vertex form: y=x^(2)-3x-5 Answer: y=

Complete the square to re-write the quadratic function in vertex form:\newliney=x2−3x−5 y=x^{2}-3 x-5 \newlineAnswer: y= y=

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Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. Complete the square to re-write the quadratic function in vertex form:\newliney=x2−3x−5 y=x^{2}-3 x-5 \newlineAnswer: y= y=
  1. Identify Coefficient and Square: To complete the square, we need to form a perfect square trinomial from the quadratic and linear terms of the function y=x2−3x−5y = x^2 - 3x - 5. We start by identifying the coefficient of the xx term, which is −3-3, and then divide it by 22 and square the result to find the constant term that will complete the square.\newlineCoefficient of xx term: −3-3\newline(−3/2)2=(3/2)2=9/4(-3/2)^2 = (3/2)^2 = 9/4
  2. Add/Subtract Constant Term: Next, we add and subtract this constant term 94\frac{9}{4} inside the function to complete the square, ensuring that the overall value of the function does not change.\newliney=x2−3x+(94)−(94)−5y = x^2 - 3x + \left(\frac{9}{4}\right) - \left(\frac{9}{4}\right) - 5
  3. Group and Combine Constants: Now, we group the perfect square trinomial and combine the constants outside the square. \newliney=(x2−3x+94)−94−5y = (x^2 - 3x + \frac{9}{4}) - \frac{9}{4} - 5
  4. Factor Perfect Square Trinomial: We can now factor the perfect square trinomial into (x−32)2(x - \frac{3}{2})^2.\newliney = (x−32)2−94−5(x - \frac{3}{2})^2 - \frac{9}{4} - 5
  5. Combine Constants with Common Denominator: To combine the constants, we need a common denominator. The common denominator for 94\frac{9}{4} and 55 is 44. We convert 55 to 204\frac{20}{4} to combine the constants.\newliney=(x−32)2−94−204y = (x - \frac{3}{2})^2 - \frac{9}{4} - \frac{20}{4}
  6. Final Result: Now, we combine the constants −94-\frac{9}{4} and −204-\frac{20}{4} to get −294-\frac{29}{4}. \newliney=(x−32)2−294y = (x - \frac{3}{2})^2 - \frac{29}{4}

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