Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation

Calculate the discriminant $b^2-4ac$ for each equation and state the expected type of solutions.

1. ⓐ

   $x^2+4x+4=0$

   $b^2-4ac={\left(4\right)}^2-4\left(1\right)\left(4\right)=0.$ There will be one rational double solution.

2. ⓑ

   $8x^2+14x+3=0$

   $b^2-4ac={\left(14\right)}^2-4\left(8\right)\left(3\right)=100.$ As $100$ is a perfect square, there will be two rational solutions.

3. ⓒ

   $3x^2-5x-2=0$

   $b^2-4ac={\left(\mathrm{-5}\right)}^2-4\left(3\right)\left(\mathrm{-2}\right)=49.$ As $49$ is a perfect square, there will be two rational solutions.

4. ⓓ

   $3x^2\mathrm{-10}x+15=0$

   $b^2-4ac={\left(\mathrm{-10}\right)}^2-4\left(3\right)\left(15\right)=\mathrm{-80.}$ There will be two complex solutions.