Lesson 17

Completing the Square and Complex Solutions

  • Let’s find complex solutions to quadratic equations by completing the square.

Problem 1

Find the solution or solutions to each equation.

  1. \(x^2+0.5x-14=0\)
  2. \(x^2+12x+36=0\)
  3. \(x^2-3x+8=0\)
  4. \(x^2+4=0\)

Problem 2

Which describes the solutions to the equation \(x^2+7=0\)?

A:

One real solution

B:

Two real solutions

C:

One non-real solution

D:

Two non-real solutions

Problem 3

Explain how you know \(\sqrt{3x+2}=\text-16\) has no solutions.

(From Unit 3, Lesson 7.)

Problem 4

Determine the number of real solutions and non-real solutions to each equation. Use the graphs; don't do any calculations to find the solutions.

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

\(y=x^2-6x+7\)

Parabola y= x squared -6x +7.

\(y=3x^2+2x+1\)

Parabola y= 3 x squared +2x + 1.

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

Parabola y= - x squared -3x +2.

Problem 5

  1. Write \((5-5i)^2\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers.
  2. Write \((5-5i)^4\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers.
(From Unit 3, Lesson 14.)

Problem 6

What number \(n\) makes this equation true?

\(x^2+11x+\frac{121}{4} = (x+n)^2\)

A:

\(\frac{11}{4}\)

B:

\(\frac{11}{2}\)

C:

11

D:

\(\frac{121}{4}\)

(From Unit 3, Lesson 16.)