Question

assignment
solve each equation by completing the square.

1. $n^2 - 14n + 63 = 0$
2. $x^2 - 8x + 14 = 0$
3. $b^2 - 18b + 45 = 0$
4. $r^2 - 20r - 60 = 0$
5. $n^2 + 12n + 42 = 0$
6. $b^2 + 8b + 43 = 0$
7. $b^2 - 4b - 67 = 0$
8. $r^2 - 12r + 100 = 0$
9. $b^2 + 10b - 15 = 0$
10. $r^2 + 14r + 88 = 0$

Explanation:

Step1: Solve \( n^2 - 14n + 63 = 0 \)

Move the constant term to the right: \( n^2 - 14n = -63 \)
Take half of -14, square it: \( (\frac{-14}{2})^2 = 49 \)
Add to both sides: \( n^2 - 14n + 49 = -63 + 49 \)
Factor left: \( (n - 7)^2 = -14 \)
Since the square of a real number can't be negative, no real solutions. If complex: \( n - 7 = \pm\sqrt{14}i \), so \( n = 7 \pm\sqrt{14}i \)

Step2: Solve \( x^2 - 8x + 14 = 0 \)

Move constant: \( x^2 - 8x = -14 \)
Half of -8 squared: \( (-4)^2 = 16 \)
Add: \( x^2 - 8x + 16 = -14 + 16 \)
Factor: \( (x - 4)^2 = 2 \)
Take square roots: \( x - 4 = \pm\sqrt{2} \)
Solve: \( x = 4 \pm\sqrt{2} \)

Step3: Solve \( b^2 - 18b + 45 = 0 \)

Move constant: \( b^2 - 18b = -45 \)
Half of -18 squared: \( (-9)^2 = 81 \)
Add: \( b^2 - 18b + 81 = -45 + 81 \)
Factor: \( (b - 9)^2 = 36 \)
Square roots: \( b - 9 = \pm6 \)
Solve: \( b = 9 + 6 = 15 \) or \( b = 9 - 6 = 3 \)

Step4: Solve \( r^2 - 20r - 60 = 0 \)

Move constant: \( r^2 - 20r = 60 \)
Half of -20 squared: \( (-10)^2 = 100 \)
Add: \( r^2 - 20r + 100 = 60 + 100 \)
Factor: \( (r - 10)^2 = 160 \)
Square roots: \( r - 10 = \pm\sqrt{160} = \pm4\sqrt{10} \)
Solve: \( r = 10 \pm4\sqrt{10} \)

Step5: Solve \( n^2 + 12n + 42 = 0 \)

Move constant: \( n^2 + 12n = -42 \)
Half of 12 squared: \( 6^2 = 36 \)
Add: \( n^2 + 12n + 36 = -42 + 36 \)
Factor: \( (n + 6)^2 = -6 \)
Complex solutions: \( n + 6 = \pm\sqrt{6}i \), so \( n = -6 \pm\sqrt{6}i \)

Step6: Solve \( b^2 + 8b + 43 = 0 \)

Move constant: \( b^2 + 8b = -43 \)
Half of 8 squared: \( 4^2 = 16 \)
Add: \( b^2 + 8b + 16 = -43 + 16 \)
Factor: \( (b + 4)^2 = -27 \)
Complex solutions: \( b + 4 = \pm3\sqrt{3}i \), so \( b = -4 \pm3\sqrt{3}i \)

Step7: Solve \( b^2 - 4b - 67 = 0 \)

Move constant: \( b^2 - 4b = 67 \)
Half of -4 squared: \( (-2)^2 = 4 \)
Add: \( b^2 - 4b + 4 = 67 + 4 \)
Factor: \( (b - 2)^2 = 71 \)
Square roots: \( b - 2 = \pm\sqrt{71} \)
Solve: \( b = 2 \pm\sqrt{71} \)

Step8: Solve \( r^2 - 12r + 100 = 0 \)

Move constant: \( r^2 - 12r = -100 \)
Half of -12 squared: \( (-6)^2 = 36 \)
Add: \( r^2 - 12r + 36 = -100 + 36 \)
Factor: \( (r - 6)^2 = -64 \)
Complex solutions: \( r - 6 = \pm8i \), so \( r = 6 \pm8i \)

Step9: Solve \( b^2 + 10b - 15 = 0 \)

Move constant: \( b^2 + 10b = 15 \)
Half of 10 squared: \( 5^2 = 25 \)
Add: \( b^2 + 10b + 25 = 15 + 25 \)
Factor: \( (b + 5)^2 = 40 \)
Square roots: \( b + 5 = \pm\sqrt{40} = \pm2\sqrt{10} \)
Solve: \( b = -5 \pm2\sqrt{10} \)

Step10: Solve \( r^2 + 14r + 88 = 0 \)

Move constant: \( r^2 + 14r = -88 \)
Half of 14 squared: \( 7^2 = 49 \)
Add: \( r^2 + 14r + 49 = -88 + 49 \)
Factor: \( (r + 7)^2 = -39 \)
Complex solutions: \( r + 7 = \pm\sqrt{39}i \), so \( r = -7 \pm\sqrt{39}i \)

Answer:

1. \( n = 7 \pm\sqrt{14}i \) (or no real solutions)
2. \( x = 4 \pm\sqrt{2} \)
3. \( b = 3 \) or \( b = 15 \)
4. \( r = 10 \pm4\sqrt{10} \)
5. \( n = -6 \pm\sqrt{6}i \) (or no real solutions)
6. \( b = -4 \pm3\sqrt{3}i \) (or no real solutions)
7. \( b = 2 \pm\sqrt{71} \)
8. \( r = 6 \pm8i \) (or no real solutions)
9. \( b = -5 \pm2\sqrt{10} \)
10. \( r = -7 \pm\sqrt{39}i \) (or no real solutions)