Problems

1. (a). Solve $(x+5)^2-49=0$, where $x$ is a real number. Simplify your answer as much as possible.

   (b). Solve $(u-7)^2-72=0$, where $u$ is a real number. Simplify your answer as much as possible.

2. (a). Solve the quadratic equation by completing the square.

    $x^2-2x-5=0$

   (b). Solve the quadratic equation by completing the square.

    $x^2+14x+35=0$

3. Use the quadratic formula to solve for $x$

    (a). $9x^2+x-2=0$

    (b). $2x^2-6x+3=0$

4. (a) Use the quadratic formula to solve for $x$.

     $7x^2=3x+2$

   Round your answer to the nearest hundredth.

   (b) Use the quadratic formula to solve for $x$.

     $2x^2+9x=-5$

   Round your answer to the nearest hundredth.

5. (a). Find all complex solutions of

       $5x^2-4x+3=0$

   (b). Find all complex solutions of

       $13x^2+5x+1=0$

6. Compute the value of the discriminant and give the number of real solutions of the following quadratic equations.

   (a). $3x^2-8x+3=0$

   (b). $-3x^2+2x-4=0$

   (c). $4x^2+20x+25=0$

Answer key

1. (a). $2, -12$

   (b). $7+6\sqrt{2}\:, 7-6\sqrt{2}$

2. (a). $1+\sqrt{6}\:,1-\sqrt{6}$

   (b). $-7+\sqrt{14}\:,-7-\sqrt{14}$

3. (a). $\dfrac{-1+\sqrt{73}}{18}\:, \dfrac{-1-\sqrt{73}}{18}$

   (b). $\dfrac{3+\sqrt{3}}{2}\:, \dfrac{3-\sqrt{3}}{2}$

4. (a). $0.79\:, -0.36$

   (b). $-0.65\:, -3.85$

5. (a). $\dfrac{2}{5}+\dfrac{\sqrt{11}}{5}i \:,\: \dfrac{2}{5}-\dfrac{\sqrt{11}}{5}i$

   (b). $-\dfrac{5}{26}+\dfrac{3\sqrt{3}}{26}i \:,\: -\dfrac{5}{26}-\dfrac{3\sqrt{3}}{26}i$

6. (a) Discriminant: $28$; Number of real solutions: $2$

   (b) Discriminant: $-44$; Number of real solutions: $0$

   (c) Discriminant: $0$; Number of real solutions: $1$