Question

solve the following quadratic equations. remember, you are to use completing the square to solve these equations. each answer must be in set notation (i.e. use the fancy brackets).
show your work!!!!

1. $k^2 + 12k - 34 = 0$

${-6 - sqrt{70}, -6 + sqrt{70}}$

1. $a^2 + 8a + 62 = 6$
2. $n^2 + 2n - 75 = 7$
3. $x^2 - 8x - 45 = 0$

Explanation:

Step1: Isolate constant (Q7)

$a^2 + 8a = 6 - 62$
$a^2 + 8a = -56$

Step2: Complete the square (Q7)

$a^2 + 8a + 16 = -56 + 16$
$(a+4)^2 = -40$

Step3: Solve for $a$ (Q7)

$a+4 = \pm\sqrt{-40} = \pm2i\sqrt{10}$
$a = -4 \pm 2i\sqrt{10}$

Step1: Isolate constant (Q8)

$n^2 + 2n = 7 + 75$
$n^2 + 2n = 82$

Step2: Complete the square (Q8)

$n^2 + 2n + 1 = 82 + 1$
$(n+1)^2 = 83$

Step3: Solve for $n$ (Q8)

$n+1 = \pm\sqrt{83}$
$n = -1 \pm \sqrt{83}$

Step1: Isolate constant (Q9)

$x^2 - 8x = 45$

Step2: Complete the square (Q9)

$x^2 - 8x + 16 = 45 + 16$
$(x-4)^2 = 61$

Step3: Solve for $x$ (Q9)

$x-4 = \pm\sqrt{61}$
$x = 4 \pm \sqrt{61}$

Answer:

1. $\{-4 - 2i\sqrt{10}, -4 + 2i\sqrt{10}\}$
2. $\{-1 - \sqrt{83}, -1 + \sqrt{83}\}$
3. $\{4 - \sqrt{61}, 4 + \sqrt{61}\}$