Home / Courses / Algebra 2 / Unit 2

Algebra 2 Unit 2: Quadratics & complex numbers

This unit finishes quadratics and introduces complex numbers, which let every quadratic have solutions.

Covers: Vertex form and completing the square, quadratic formula, discriminant, imaginary unit i, complex arithmetic

Study guide

Quadratics can be written three ways: standard form ax² + bx + c, vertex form a(x − h)² + k with vertex (h, k), and factored form a(x − r)(x − s) with zeros r and s. Completing the square converts standard form to vertex form: factor out a, take half the x-coefficient, square it, add and subtract it, and rewrite the perfect square.

The quadratic formula, x = [−b ± √(b² − 4ac)]/(2a), solves any quadratic. The discriminant b² − 4ac predicts the solutions: positive gives two real roots, zero gives one repeated real root, and negative gives two complex roots. The sum of the roots is −b/a and the product is c/a.

When the discriminant is negative, solutions involve the imaginary unit i, defined by i² = −1, so √−9 = 3i. A complex number has the form a + bi with real part a and imaginary part b. Add and subtract complex numbers by combining real and imaginary parts. Multiply by distributing and replacing i² with −1. Powers of i repeat every four: i, −1, −i, 1.

The conjugate of a + bi is a − bi, and their product a² + b² is real. To divide complex numbers, multiply the top and bottom by the conjugate of the denominator. Complex roots of real quadratics always come in conjugate pairs.

To solve a system with a line and a circle or parabola, solve the linear equation for one variable, substitute into the other equation, and solve the resulting quadratic. Each real solution gives one intersection point, so there can be 0, 1 or 2, and you find each y-value by substituting back into the line. To square a + bi, use (a + bi)² = a² + 2abi + b²i² = (a² − b²) + 2abi. The common error is forgetting that b²i² = −b².

Key ideas

• Complete the square to convert to vertex form.
• The discriminant b² − 4ac predicts the type and number of roots.
• i = √−1, and powers of i cycle: i, −1, −i, 1.
• Add complex numbers by combining real and imaginary parts.
• Divide by multiplying top and bottom by the conjugate.

Formulas and rules

Common mistakes

• Writing √−9 as −3 instead of 3i.
• Forgetting i² = −1 when multiplying.

Words to know

i²

−1

Powers of i

They cycle every 4: i, −1, −i, 1.

Complex conjugate

a + bi and a − bi; their product is a² + b².

Negative discriminant

Two complex (non-real) roots.

Sum of the roots of ax² + bx + c

−b/a

Vertex form

a(x − h)² + k, with vertex (h, k).

Completing the square

Rewriting x² + bx as (x + b/2)² − (b/2)².

Quadratic formula

x = [−b ± √(b² − 4ac)]/(2a)

Imaginary unit (i)

i = √−1, so i² = −1.

Complex number

a + bi, with real part a and imaginary part b.

Product of the roots

c/a for ax² + bx + c.

Dividing complex numbers

Multiply top and bottom by the conjugate of the denominator.

Sample practice questions

41 more questions like these, from level 1 to 5, are in the practice set.