Lesson 19

Real and Non-Real Solutions

Let's create and solve quadratic equations.

Problem 1

Without calculating the solutions, determine whether each equation has real solutions or not.

  1. \(\text-0.5x^2+3x=0\)
  2. \(x^2-4x+7=0\)
  3. \(2x^2-2x-1=0\)
  4. \(\text-0.5x^2+3x=3\)
  5. \(x^2-4x+7=5\)
  6. \(2x^2-2x-1=\text-4\)

\(y = \text-0.5x^2+3x\)

Parabola, Y= -0 point 5 x squared + 3x.

\(y = x^2-4x+7\)

Parabola, Y= X squared -4x +7

\(y = 2x^2-2x-1\)

Parabola y= 2x squared -2x -1

Problem 2

The graph shows the equation \(y=2x^2+0.5x-4\).

Based on the graph, what number could you put in the box to create an equation that has no real solutions?
\(\displaystyle 2x^2+0.5x-4 = \boxed{\phantom{30}}\)

Problem 3

The graph shows the equation \(y = 1.5x^2-3x+2\).

Parabola facing up. Vertex=1 comma one half. No roots. X intercept =2.
  1. Without calculating the solutions, determine whether \(1.5x^2-3x+2=0\) has real solutions.
  2. Show how to solve \(1.5x^2-3x+2=0\).

Problem 4

Write a quadratic equation that has two non-real solutions. How did you decide what equation to write?

Problem 5

Find the solution or solutions to each equation.

  1. \(\text-2x^2+2x=2.5\)
  2. \(4.5x^2+3x+\frac12=0\)
  3. \(\frac12 x^2+5x=\text-14\)
  4. \(\text- x^2 -1.5x + 5 = 7\)

Problem 6

Elena and Kiran were solving the equation \(2x^2-4x+3=0\) and they got different answers. Elena wrote \(1 \pm i\sqrt{0.5}\), and Kiran wrote \(1 \pm \frac{i\sqrt8}{4}\). Are their answers equivalent? Say how you know.