DC CAPE Algebra II Complex Numbers and Quadratic Equations. Practice it free.

This domain addresses the complex number system and quadratic reasoning. This Algebra II reporting domain maps to 7 practice skills and 8 representative questions from the playable bank.

Algebra IIComplex Numbers and Quadratic Equations7 mapped skills
What the test measures

Complex Numbers and Quadratic Equations skills

  1. perform arithmetic operations with complex numbers

  2. use the complex number system to solve quadratic equations

  3. solve quadratics by inspection, taking square roots, completing the square, and the quadratic formula

Standards basis

Common Core State Standards (CCSS-M) — District of Columbia

How the DC CAPE reports it

DC CAPE math continues DC’s PARCC-era alignment to the Common Core State Standards and uses the same general development/review approach described by OSSE. The page says DC Math follows the same rigorous development and review process and measures the knowledge and skills that matter most for DC students, while also explicitly tying the assessments to CCSS.

Blueprint & weighting

DC CAPE uses the PARCC-derived / Common Core-based framework (by grade domains structure).

Practice by skill

Topics mapped to this domain

Discriminant analysis

Grade 915 questions

The discriminant and quadratic formula

Grade 920 questions

Quadratic distinct real solutions count

Grade 910 questions

Solving quadratics (leading coefficient not 1)

Grade 910 questions

Multiplying and squaring complex numbers (ACT)

Grade 1110 questions

Complex number operations

Grade 1117 questions

Rewriting complex numbers

Grade 1110 questions
Try it now

8 free questions · 0/0 correct

Practice playeasy

How many distinct real roots does x2−25=0x^{2} - 25 = 0 have?

Here a=1a = 1, b=0b = 0, and c=−25c = -25. Substituting into b2−4acb^{2} - 4ac gives 02−4(1)(−25)=100,0^{2} - 4(1)(-25) = 100\textsf{,} so there are 22 distinct real roots.

Practice playeasy

How many distinct real solutions does x2+6x+9=0x^{2} + 6x + 9 = 0 have?

Here a=1a = 1, b=6b = 6, and c=9c = 9. Substituting into b2−4acb^{2} - 4ac gives 62−4(1)(9)=06^{2} - 4(1)(9) = 0, so there is exactly 11 distinct real solution.

Practice playmedium

How many distinct real solutions does the equation 3(x+2)2+15=03(x + 2)^{2} + 15 = 0 have?

Subtract 1515: 3(x+2)2=−15.3(x + 2)^{2} = -15\textsf{.} Divide by 33: (x+2)2=−5.(x + 2)^{2} = -5\textsf{.} A real number squared cannot be negative, so there are 00 distinct real solutions.

Practice playeasy

What is the positive solution of 2x2−x−15=0?2x^{2} - x - 15 = 0\textsf{?}

The quadratic factors as (2x+5)(x−3)=0.(2x + 5)(x - 3) = 0\textsf{.} By the zero-product property, 2x+5=02x + 5 = 0 or x−3=0,x - 3 = 0\textsf{,} so x=−52x = -\dfrac{5}{2} or x=3.x = 3\textsf{.} The positive solution is 3.3\textsf{.}

Practice playhard

(2+5i)2=(2+5i)^{2}=

Expand: (2+5i)2=4+20i+25i2(2+5i)^{2}=4+20i+25i^{2}. Since i2=−1i^{2}=-1, this is 4+20i−25=−21+20i.4+20i-25=-21+20i\textsf{.}

Practice playeasy

What is (2i)2(2i)^2?

(2i)2=4i2=4(−1)=−4(2i)^2 = 4i^2 = 4(-1) = -4.

Practice playeasy

Given i=−1i = \sqrt{-1}, what is 81+−25\sqrt{81} + \sqrt{-25}?

81=9\sqrt{81} = 9 and −25=5i\sqrt{-25} = 5i. So 81+−25=9+5i\sqrt{81} + \sqrt{-25} = 9 + 5i.

Practice playeasy

How many distinct real roots does x2+1=0x^{2} + 1 = 0 have?

Here a=1a = 1, b=0b = 0, and c=1c = 1. Substituting into b2−4acb^{2} - 4ac gives 02−4(1)(1)=−4,0^{2} - 4(1)(1) = -4\textsf{,} so there are 00 distinct real roots.

Keep practicing

Turn complex numbers and quadratic equations into game time.

The DC CAPE placement starts with this test's real coverage map and finds the right difficulty.

Batting Cage game coverBatting CagePlay →Blossom Leap game coverBlossom LeapPlay →Buzzer Beater game coverBuzzer BeaterPlay →Minecart Junction game coverMinecart JunctionPlay →
Related Common Core standards
HSA.REI.B.4HSN.CN.A.2HSN.CN.A.1
Source and test overview
All DC CAPE prepOfficial test source
Reporting categories

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