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Quadratics with Complex Solutions

  • Grade 12
  • Mathematics
  • N-CN.C.7
  • Number & Quantity
  • Free lesson

Aligned to N-CN.C.7 — Common Core State Standards for Mathematics.

What this lesson teaches

When the discriminant b^2 − 4ac is negative, the quadratic formula produces complex solutions containing i. These solutions always come as a conjugate pair.

Worked examples

Example 1

Solve x^2 − 2x + 5 = 0. Here x = (2 ± √(4 − 20))/2 = (2 ± √(−16))/2 = (2 ± 4i)/2 = 1 ± 2i.

Example 2

Solve 2x² + 2x + 5 = 0. Here a = 2, b = 2, c = 5, so b² − 4ac = 4 − 40 = −36. Then x = (−2 ± √(−36))/4 = (−2 ± 6i)/4 = −1/2 ± (3/2)i. Note the two roots are conjugates — that always happens when a quadratic has real coefficients.

Example 3

Common mistake: forgetting to divide BOTH parts by 2a. From x = (4 ± 6i)/2 the answer is 2 ± 3i, not 4 ± 3i and not 2 ± 6i — the denominator divides the real part and the imaginary part alike. Split the fraction explicitly, 4/2 ± (6/2)i, and neither half gets skipped.

Practice questions

  1. Solve x^2 + 9 = 0.
  2. Solve x^2 + 1 = 0.
  3. Solve x^2 − 4x + 13 = 0 using the quadratic formula.

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Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.

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En español

Cuadráticas con soluciones complejas

Cuando el discriminante b^2 − 4ac es negativo, la fórmula cuadrática produce soluciones complejas que contienen i. Estas soluciones siempre aparecen como un par conjugado.

Ejemplo 1: Resuelve x^2 − 2x + 5 = 0. Aquí x = (2 ± √(4 − 20))/2 = (2 ± √(−16))/2 = (2 ± 4i)/2 = 1 ± 2i.

Ejemplo 2: Resuelve 2x² + 2x + 5 = 0. Aquí a = 2, b = 2, c = 5, así que b² − 4ac = 4 − 40 = −36. Entonces x = (−2 ± √(−36))/4 = (−2 ± 6i)/4 = −1/2 ± (3/2)i. Nota que las dos raíces son conjugadas — eso siempre ocurre cuando una cuadrática tiene coeficientes reales.

Ejemplo 3: Error común: olvidar dividir AMBAS partes entre 2a. De x = (4 ± 6i)/2 la respuesta es 2 ± 3i, no 4 ± 3i ni 2 ± 6i — el denominador divide por igual la parte real y la imaginaria. Separa la fracción explícitamente, 4/2 ± (6/2)i, y así no se salta ninguna mitad.

More Number & Quantity lessons

Adding and Multiplying Complex NumbersN-CN.A.2The Fundamental Theorem of AlgebraN-CN.C.9Finding Vector ComponentsN-VM.A.2Adding and Subtracting VectorsN-VM.B.4Representing Data with MatricesN-VM.C.6Matrix Addition and MultiplicationN-VM.C.8

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