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Solve x^(2)-12 x+19=0 by completing the square.

1515. Solve x2−12x+19=0 x^{2}-12 x+19=0 by completing the square.

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Full solution

Q. 1515. Solve x2−12x+19=0 x^{2}-12 x+19=0 by completing the square.
  1. Identify coefficients: Identify the quadratic and linear coefficients from the equation x2−12x+19=0x^2 - 12x + 19 = 0. Here, a=1a = 1 and b=−12b = -12.
  2. Form perfect square: Divide the coefficient of xx by 22 and square the result to form the perfect square term. (−12/2)2=36(-12/2)^2 = 36.
  3. Rewrite equation: Rewrite the equation incorporating this square. x2−12x+36−36+19=0x^2 - 12x + 36 - 36 + 19 = 0 turns into (x−6)2−17=0(x - 6)^2 - 17 = 0.
  4. Isolate perfect square: Isolate the perfect square by moving −17-17 to the right side of the equation. (x−6)2=17(x - 6)^2 = 17.
  5. Take square root: Take the square root of both sides, remembering to consider both the positive and negative roots. x−6=±17x - 6 = \pm\sqrt{17}.
  6. Solve for x: Solve for x by adding 66 to both sides. x=6±17.x = 6 \pm \sqrt{17}.

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