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A.2(B)·Algebra II·Algebraic MethodsSupporting

Complex Solutions

Use complex numbers to describe the solutions of quadratic equations.

Students simplify nonreal quadratic solutions into a±bia\pm bi form by rewriting negative square roots and dividing every numerator term.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Algebra II released practice50 minutes

Learn the concept

Introduce i

The imaginary unit satisfies i2=−1i^2=-1. For a positive real c, the principal square root is −c=ic\sqrt{-c}=i\sqrt c. Thus −49=7i\sqrt{-49}=7i.

Simplify before dividing

In an expression from the quadratic formula, simplify the radical and preserve the ±\pm symbol. The two signs give a conjugate pair when the other quantities are real.

Divide every term

Use p±qid=pd±qdi\frac{p\pm qi}{d}=\frac pd\pm\frac qd i for d≠0d\ne0. Reduce each fraction separately; the imaginary term does not escape the denominator.

Worked example

Write −18±−6412\frac{-18\pm\sqrt{-64}}{12} in standard complex form.

  1. Rewrite −64=8i\sqrt{-64}=8i.
  2. Split the fraction: −1812±812i\frac{-18}{12}\pm\frac{8}{12}i.
  3. Reduce both fractions: −32±23i-\frac32\pm\frac23i.

−32±23i-\frac32\pm\frac23i

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Solving Quadratic Equations

UT IPSI · Kid2Kid / Texas Education Agency

Connect quadratic equations to solution expressions, then extend negative discriminants using i.

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Try it yourself

TEKSForge practice · unscored

  1. 1. Simplify 14±−486\frac{14\pm\sqrt{-48}}{6}.

    Show answer and reasoning

    73±233i\frac73\pm\frac{2\sqrt3}{3}i

    −48=43i\sqrt{-48}=4\sqrt3i; divide 14 and 43i4\sqrt3i by 6.

  2. 2. Solve x2+16=0x^2+16=0.

    Show answer and reasoning

    x=±4ix=\pm4i

    x2=−16x^2=-16, so the two roots are 4i4i and −4i-4i; each squares to −16-16.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 25 · 1 point

The roots of a quadratic equation are given by the expression −25±−10025\frac{-25 \pm \sqrt{-100}}{25}. Which of the following is an equivalent expression?

  1. −1±2i5-1 \pm \frac{2i}{5}
  2. −1±2i-1 \pm 2i
  3. −25±2i5-25 \pm \frac{2i}{5}
  4. −25±2i-25 \pm 2i
Show STAAR answer and explanation

−1±2i5-1 \pm \frac{2i}{5}

−100=10i\sqrt{-100}=10i. Divide each term of the numerator by 25: −2525±10i25=−1±2i5\frac{-25}{25}\pm\frac{10i}{25}=-1\pm\frac{2i}{5}.

−1±2i-1 \pm 2i: This divides 10i10i by 5 instead of 25.

−25±2i5-25 \pm \frac{2i}{5}: The −25 must also be divided by 25.

−25±2i-25 \pm 2i: Neither term was divided by 25 correctly.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Use the definition i2=−1i^2=-1.
  • Simplify square roots of negative real numbers.
  • Reduce both terms of a complex quotient.

Instructional sequence

  1. 1

    Engage

    5 min

    Compare equations with positive and negative squared values.

  2. 2

    Direct instruction

    10 min

    Define i and simplify negative square roots.

  3. 3

    Guided practice

    13 min

    Simplify quadratic-formula expressions term by term.

  4. 4

    Compare and verify

    15 min

    Check conjugate solutions using a simple quadratic.

  5. 5

    Closure

    7 min

    Explain denominator distribution on the exit ticket.

Misconceptions to address

“A negative radicand makes the answer a negative real number.”

Teacher move: A real square cannot be negative; use i to express nonreal roots.

“The denominator divides only the real term.”

Teacher move: The fraction bar covers the entire numerator, so divide both terms.