NumeriveMATH
CurriculumFamily workspaceStudentMy characterAchievementsContact
← Intermediate Algebra levels

Intermediate Algebra / LEVEL 1 · DIFFICULTY 1/5

Complex Numbers and Quadratics

Extend arithmetic and interpret quadratic roots.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST IA 1.1Multiplying Complex NumbersRead the lesson
  2. MINI QUEST IA 1.2Modulus and ConjugationRead the lesson
  3. MINI QUEST IA 1.3DiscriminantsRead the lesson
  4. LEVEL CHECKCastle challenge

    6 questions across this level.

    Sign in for the level check →

STAGE IA 1.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Multiplying Complex Numbers

Goal: Understand and apply multiplying complex numbers.

Before you begin: Linear and quadratic equations, factoring and introductory functions.

Understand the idea

Complex numbers obey distributive multiplication with the additional rule i²=−1. Products of two imaginary terms contribute to the real part, so both constant products matter.

(a+bi)(c+di)=(ac−bd)+(ad+bc)i

Choose and carry out a method

Expand all four products, replace i² by −1, and group real and imaginary terms. Enter only the component requested.

Check the reasoning

Multiplying by i rotates a point by a quarter-turn. A second multiplication by i negates it, consistent with i²=−1.

WORKED EXAMPLE 1

Find the real part of (3+6i)(2+3i), where i²=-1.

  1. Expand the product and replace i² by -1.
  2. Real terms are 2·3+3·6·(-1).
  3. The real part is -12; the cross-terms belong to the imaginary part.

-12

WORKED EXAMPLE 2

Find the real part of (4+7i)(2+3i), where i²=-1.

  1. Expand the product and replace i² by -1.
  2. Real terms are 2·4+3·7·(-1).
  3. The real part is -13; the cross-terms belong to the imaginary part.

-13

Common pitfalls

Possible mix-up: The real part is only ac.

The product bi·di contributes −bd to the real part.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why does multiplying two imaginary terms produce a real term?

Preview the eight practice prompts
  1. Find the real part of (6+9i)(2+3i), where i²=-1.
  2. Find the real part of (7+10i)(2+3i), where i²=-1.
  3. Find the real part of (8+11i)(2+3i), where i²=-1.
  4. Find the real part of (9+12i)(2+3i), where i²=-1.
  5. Find the real part of (10+13i)(2+3i), where i²=-1.
  6. Find the real part of (11+14i)(2+3i), where i²=-1.
  7. Two planar transformations are encoded by complex factors 12+15i and 2+3i. Their composition uses the product. What is its real part? New context
  8. Two planar transformations are encoded by complex factors 13+16i and 2+3i. Their composition uses the product. What is its real part? New context
Open stage IA 1.1 in the student workspace →

STAGE IA 1.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Modulus and Conjugation

Useful preparation: Multiplying Complex Numbers

Goal: Understand and apply modulus and conjugation.

Before you begin: Multiplying Complex Numbers

Understand the idea

The modulus of x+yi is its distance from the origin in the complex plane. Multiplying a number by its conjugate eliminates the imaginary cross terms and yields the squared distance.

|x+yi|²=(x+yi)(x−yi)=x²+y²

Choose and carry out a method

Square both real coordinates and add. If the question asks for the squared modulus, stop before taking a square root.

Check the reasoning

The result is nonnegative, even for negative coordinates. Conjugation preserves distance while reflecting the point across the real axis.

WORKED EXAMPLE 1

Find |3-5i|².

  1. Multiplying a complex number by its conjugate gives its squared modulus.
  2. (3-5i)(3+5i)=3²+5².
  3. The squared distance is 34, which is real and nonnegative.

34

WORKED EXAMPLE 2

Find |4-6i|².

  1. Multiplying a complex number by its conjugate gives its squared modulus.
  2. (4-6i)(4+6i)=4²+6².
  3. The squared distance is 52, which is real and nonnegative.

52

Common pitfalls

Possible mix-up: Square the imaginary unit as part of the distance.

The coordinate y is squared; squared distance is x²+y².

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Explain geometrically why conjugate numbers have equal modulus.

Preview the eight practice prompts
  1. Find |6-8i|².
  2. Find |7-9i|².
  3. Find |8-10i|².
  4. Find |9-11i|².
  5. Find |10-12i|².
  6. Find |11-13i|².
  7. A displacement is represented by the complex number 12-14i. What is its squared distance from the origin? New context
  8. A displacement is represented by the complex number 13-15i. What is its squared distance from the origin? New context
Open stage IA 1.2 in the student workspace →

STAGE IA 1.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Discriminants

Useful preparation: Modulus and Conjugation

Goal: Understand and apply discriminants.

Before you begin: Modulus and Conjugation

Understand the idea

Completing the square in a quadratic reveals the expression b²−4ac. Its sign controls whether the two roots are distinct real numbers, one repeated real number or a conjugate nonreal pair.

Δ=b²−4ac

Choose and carry out a method

Identify a, b and c with their signs before substituting. Compute the discriminant using parentheses around a negative coefficient.

Check the reasoning

A perfect-square discriminant may give rational roots when coefficients are integers. A positive value alone guarantees real roots, not integer roots.

WORKED EXAMPLE 1

Find the discriminant of x²-5x+3=0.

  1. For ax²+bx+c=0, the discriminant is b²-4ac.
  2. D=(-5)²-4·1·3.
  3. D=13, so this example has two distinct real roots.

13

WORKED EXAMPLE 2

Find the discriminant of x²-7x+4=0.

  1. For ax²+bx+c=0, the discriminant is b²-4ac.
  2. D=(-7)²-4·1·4.
  3. D=33, so this example has two distinct real roots.

33

Common pitfalls

Possible mix-up: A negative b gives a negative b².

Squaring a signed coefficient gives a nonnegative value.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

What can the discriminant tell you without calculating either root?

Preview the eight practice prompts
  1. Find the discriminant of x²-11x+6=0.
  2. Find the discriminant of x²-13x+7=0.
  3. Find the discriminant of x²-15x+8=0.
  4. Find the discriminant of x²-17x+9=0.
  5. Find the discriminant of x²-19x+10=0.
  6. Find the discriminant of x²-21x+11=0.
  7. A projectile-height model reaches the ground when x²-23x+12=0. What is the discriminant used to check whether real crossing times exist? New context
  8. A projectile-height model reaches the ground when x²-25x+13=0. What is the discriminant used to check whether real crossing times exist? New context
Open stage IA 1.3 in the student workspace →
Browse another level →