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Solve for the roots in simplest form by completing the square:x, squared, minus, 10, x, minus, 167, equals, 0x 2 −10x−167=0

Question

Solve for the roots in simplest form by completing the square:

Given the equation:

x^2 - 10x - 167 = 0

Complete the square to find the roots.

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Solution

Sure, let's solve the quadratic equation step by step:

  1. First, let's rewrite the equation: x^2 - 10x - 167 = 0

  2. To complete the square, we need to find a number that can be added and subtracted to make the equation a perfect square trinomial. This number is found by taking half of the coefficient of x, squaring it. In this case, it's (-10/2)^2 = 25.

  3. Add and subtract this number inside the equation: x^2 - 10x + 25 - 25 - 167 = 0

  4. Now, the first three terms form a perfect square trinomial: (x - 5)^2 - 25 - 167 = 0

  5. Simplify the equation: (x - 5)^2 = 192

  6. Take the square root of both sides: x - 5 = ±sqrt(192)

  7. Simplify the square root: x - 5 = ±8sqrt(3)

  8. Finally, solve for x: x = 5 ± 8sqrt(3)

So, the roots of the equation are x = 5 + 8sqrt(3) and x = 5 - 8sqrt(3).

This problem has been solved

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