Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 19

Capitolo 5, Problema 19

In Exercises 9–20, find each product and write the result in standard form. (2 + 3i)²

Guida verificata passo dopo passo

Recall that to find the square of a complex number, such as $(2 + 3i)^2$, you can use the formula for the square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$.

Identify $a = 2$ and $b = 3i$ in the expression $(2 + 3i)^2$.

Apply the formula: calculate $a^2 = (2)^2$, $2ab = 2 \times 2 \times 3i$, and $b^2 = (3i)^2$ separately.

Remember that $i^2 = -1$, so when you calculate $b^2 = (3i)^2$, rewrite it as $3^2 \times i^2 = 9 \times (-1)$.

Combine all the terms: $a^2 + 2ab + b^2$, simplify the real and imaginary parts separately to write the result in standard form $x + yi$.

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Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form means writing the result explicitly as a sum of a real number and an imaginary number multiplied by i.

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Complex Numbers In Polar Form

Binomial Expansion (Square of a Binomial)

Squaring a binomial (x + y)² involves expanding it as x² + 2xy + y². This method is essential for multiplying complex numbers like (2 + 3i)² by treating 2 and 3i as the terms x and y.

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Solving Quadratic Equations by Completing the Square

Properties of the Imaginary Unit i

The imaginary unit i satisfies i² = -1. This property is crucial when simplifying powers of i during multiplication, converting terms like (3i)² into real numbers to write the final answer in standard form.

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Imaginary Roots with the Square Root Property

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