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Solve each equation by completing the square. 8) p^2 + 2p - 15 = 0 9) a^2 - 6a - 7 = 2 10) k^2 - 4k - 7 = 0...

Solve each equation by completing the square.
8) p^2 + 2p - 15 = 0
9) a^2 - 6a - 7 = 2
10) k^2 - 4k - 7 = 0
11) m^2 - 8m - 9 = 0
12) n^2 - 8n + 20 = 0
13) p^2 + 2p + 17 = 0
14) x^2 + 4x + 5 = 0
15) x^2 - 8x + 23 = 0

Answer

  1. To solve p^2 + 2p - 15 = 0 by completing the square:

    1. Move the constant to the other side: p^2 + 2p = 15.

    2. Take half of the coefficient of p, square it, and add to both sides: (2/2)^2 = 1, so p^2 + 2p + 1 = 16.

    3. Rewrite the left side as a square: (p + 1)^2 = 16.

    4. Solve for p: p + 1 = ±4, so p = 3 or p = -5.

  2. To solve a^2 - 6a - 7 = 2 by completing the square:

    1. Move the constant to the other side: a^2 - 6a = 9.

    2. Take half of the coefficient of a, square it, and add to both sides: (-6/2)^2 = 9, so a^2 - 6a + 9 = 18.

    3. Rewrite the left side as a square: (a - 3)^2 = 18.

    4. Solve for a: a - 3 = ±√18, so a = 3 ± 3√2.

  3. To solve k^2 - 4k - 7 = 0 by completing the square:

    1. Move the constant to the other side: k^2 - 4k = 7.

    2. Take half of the coefficient of k, square it, and add to both sides: (-4/2)^2 = 4, so k^2 - 4k + 4 = 11.

    3. Rewrite the left side as a square: (k - 2)^2 = 11.

    4. Solve for k: k - 2 = ±√11, so k = 2 ± √11.

  4. To solve m^2 - 8m - 9 = 0 by completing the square:

    1. Move the constant to the other side: m^2 - 8m = 9.

    2. Take half of the coefficient of m, square it, and add to both sides: (-8/2)^2 = 16, so m^2 - 8m + 16 = 25.

    3. Rewrite the left side as a square: (m - 4)^2 = 25.

    4. Solve for m: m - 4 = ±5, so m = 9 or m = -1.

  5. To solve n^2 - 8n + 20 = 0 by completing the square:

    1. Move the constant to the other side: n^2 - 8n = -20.

    2. Take half of the coefficient of n, square it, and add to both sides: (-8/2)^2 = 16, so n^2 - 8n + 16 = -4.

    3. Rewrite the left side as a square: (n - 4)^2 = -4.

    4. Solve for n: n - 4 = ±2i, so n = 4 ± 2i.

  6. To solve p^2 + 2p + 17 = 0 by completing the square:

    1. Move the constant to the other side: p^2 + 2p = -17.

    2. Take half of the coefficient of p, square it, and add to both sides: (2/2)^2 = 1, so p^2 + 2p + 1 = -16.

    3. Rewrite the left side as a square: (p + 1)^2 = -16.

    4. Solve for p: p + 1 = ±4i, so p = -1 ± 4i.

  7. To solve x^2 + 4x + 5 = 0 by completing the square:

    1. Move the constant to the other side: x^2 + 4x = -5.

    2. Take half of the coefficient of x, square it, and add to both sides: (4/2)^2 = 4, so x^2 + 4x + 4 = -1.

    3. Rewrite the left side as a square: (x + 2)^2 = -1.

    4. Solve for x: x + 2 = ±i, so x = -2 ± i.

  8. To solve x^2 - 8x + 23 = 0 by completing the square:

    1. Move the constant to the other side: x^2 - 8x = -23.

    2. Take half of the coefficient of x, square it, and add to both sides: (-8/2)^2 = 16, so x^2 - 8x + 16 = -7.

    3. Rewrite the left side as a square: (x - 4)^2 = -7.

    4. Solve for x: x - 4 = ±√7i, so x = 4 ± √7i.

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