Solutions to Problem Set 25

Math 101-06

11-10-2017

[Completing the square]

In these problems, complex number solutions are okay.

1. Solve the quadratic equation:

$$x^2 = 20.$$

$$\eqalign{ x^2 & = 20 \cr x & = \pm \sqrt{20} \cr x & = \pm 2 \sqrt{5} \quad\halmos \cr}$$


2. Solve the quadratic equation:

$$x^2 = -36.$$

$$\eqalign{ x^2 & = -36 \cr x & = \pm \sqrt{-36} \cr x & = \pm 6 i \quad\halmos \cr}$$


3. Solve the quadratic equation:

$$(x - 8)^2 = 11.$$

$$\eqalign{ (x - 8)^2 & = 11 \cr x - 8 & = \pm \sqrt{11} \cr x & = 8 \pm \sqrt{11} \quad\halmos \cr}$$


4. Solve the quadratic equation:

$$(x + 10)^2 = -16.$$

$$\eqalign{ (x + 10)^2 & = -16 \cr x + 10 & = \sqrt{-16} \cr x & = -10 \pm 4 i \quad\halmos \cr}$$


5. Solve the quadratic equation by completing the square:

$$x^2 + 10 x = 8.$$

Since $\dfrac{1}{2} \cdot 10 =
   5$ and $5^2 = 25$ , I need to add 25 to both sides:

$$\eqalign{ x^2 + 10 x & = 8 \cr x^2 + 10 x + 25 & = 8 + 25 \cr (x + 5)^2 & = 33 \cr x + 5 & = \pm \sqrt{33} \cr x & = -5 \pm \sqrt{33} \quad\halmos \cr}$$


6. Solve the quadratic equation by completing the square:

$$x^2 + 8 x = 8.$$

Since $\dfrac{1}{2} \cdot 8 =
   4$ and $4^2 = 16$ , I need to add 16 to both sides:

$$\eqalign{ x^2 + 8 x & = 8 \cr x^2 + 8 x + 16 & = 8 + 16 \cr (x + 4)^2 & = 24 \cr x + 4 & = \pm \sqrt{24} \cr x & = -4 \pm 2 \sqrt{6} \quad\halmos \cr}$$


7. Solve the quadratic equation by completing the square:

$$x^2 + 6 x = -21.$$

Since $\dfrac{1}{2} \cdot 6 =
   3$ and $3^2 = 9$ , I need to add 9 to both sides:

$$\eqalign{ x^2 + 6 x & = -21 \cr x^2 + 6 x + 9 & = -21 + 9 \cr (x + 3)^2 & = -12 \cr x + 3 & = \pm \sqrt{-12} = 2 i \sqrt{3} \cr x & = -3 \pm 2 i \sqrt{3} \quad\halmos \cr}$$


8. Solve the quadratic equation by completing the square:

$$x^2 - 8 x = -20.$$

Since $\dfrac{1}{2} \cdot (-8) =
   -4$ and $(-4)^2 = 16$ , I need to add 16 to both sides:

$$\eqalign{ x^2 - 8 x & = -20 \cr x^2 - 8 x + 16 & = -20 + 16 \cr (x - 4)^2 & = -4 \cr x - 4 & = \pm \sqrt{-4} \cr x & = 4 \pm 2 i \quad\halmos \cr}$$


9. Solve the quadratic equation by completing the square:

$$x^2 - 12 x = 4.$$

Since $\dfrac{1}{2} \cdot (12) =
   -6$ and $(-6)^2 = 36$ , I need to add 36 to both sides:

$$\eqalign{ x^2 - 12 x & = 4 \cr x^2 - 12 x + 36 & = 4 + 36 \cr (x - 6)^2 & = 40 \cr x - 6 & = \pm \sqrt{40} \cr x & = 6 \pm 2 \sqrt{10} \quad\halmos \cr}$$


10. Solve the quadratic equation by completing the square:

$$x^2 + 10 x = -43.$$

Since $\dfrac{1}{2} \cdot 10 =
   5$ and $5^2 = 25$ , I need to add 25 to both sides:

$$\eqalign{ x^2 + 10 x & = -43 \cr x^2 + 10 x + 25 & = -43 + 25 \cr (x + 5)^2 & = -18 \cr x + 5 & = \pm \sqrt{-18} \cr x & = -5 \pm 3 i \sqrt{2} \quad\halmos \cr}$$


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