# 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.

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

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

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

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

### 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 \(y = 1.5x^2-3x+2\).

- Without calculating the solutions, determine whether \(1.5x^2-3x+2=0\) has real solutions.
- 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.

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

### 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.