Circuit Breaker Worksheet
Circuit Breaker
Quadratic Formula: Complex Solutions
Name:
Date:
Tech Refresher
If the discriminant \( b^2 - 4ac \) is negative, solutions are complex. Express your final answers in standard form: \( a \pm bi \).
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
01 Discriminant Diagnostics
1.1 \( x^2 + 2x + 5 = 0 \)
D =
Complex?
1.2 \( 3x^2 - x + 2 = 0 \)
D =
Complex?
02 Full System Reboot
Solve using the quadratic formula. Show all work. Express in simplified \( a \pm bi \) form.
Circuit #2.1 x² - 6x + 13 = 0
Step-by-Step Work:
Final Solution:
Circuit #2.2 x² + 4x + 29 = 0
Step-by-Step Work:
Final Solution:
Circuit #2.3 2x² - 2x + 5 = 0
Step-by-Step Work:
Final Solution:
Circuit #2.4 4x² + 8x + 7 = 0
Step-by-Step Work:
Final Solution:
Circuit #2.5 3x² + 12 = 0
Step-by-Step Work:
Final Solution:
Negative Discriminant // Simplified Radical // Standard Complex Form
Circuit Breaker Answer Key
Answer Key
Circuit Breaker: Complex Quadratics
Teacher Reference
01 // Discriminant Diagnostics
1.1 \( x^2 + 2x + 5 = 0 \)
\( D = 2^2 - 4(1)(5) = -16 \)
Complex solutions
1.2 \( 3x^2 - x + 2 = 0 \)
\( D = (-1)^2 - 4(3)(2) = -23 \)
Complex solutions
02 // Procedural Solutions
2.1
\( x^2 - 6x + 13 = 0 \)
\( x = \frac{6 \pm \sqrt{-16}}{2} = \frac{6 \pm 4i}{2} \)
\( x = 3 \pm 2i \)
2.2
\( x^2 + 4x + 29 = 0 \)
\( x = \frac{-4 \pm \sqrt{-100}}{2} = \frac{-4 \pm 10i}{2} \)
\( x = -2 \pm 5i \)
2.3
\( 2x^2 - 2x + 5 = 0 \)
\( x = \frac{2 \pm \sqrt{4 - 40}}{4} = \frac{2 \pm 6i}{4} \)
\( x = \frac{1}{2} \pm \frac{3}{2}i \)
2.4
\( 4x^2 + 8x + 7 = 0 \)
\( x = \frac{-8 \pm \sqrt{-48}}{8} = \frac{-8 \pm 4i\sqrt{3}}{8} \)
\( x = -1 \pm \frac{\sqrt{3}}{2}i \)
2.5
\( 3x^2 + 12 = 0 \)
\( 3x^2 = -12 \rightarrow x^2 = -4 \)
\( x = \pm 2i \)
Solutions verified using the Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
