Complex numbers are a fundamental topic on the ACT Math test, appearing in questions about algebra, polynomial equations, and advanced problem-solving. Understanding how to work with imaginary numbers, complex conjugates, powers of i, multiplication, division, and quadratic equations will help you solve challenging ACT Math questions with confidence.
In this 2026 ACT Math study guide, you’ll learn the essential rules for complex numbers, explore step-by-step examples, and practice solving ACT-style questions. Whether you’re reviewing the basics or strengthening advanced skills, this guide will help you improve accuracy and boost your ACT Math score.
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When raising complex numbers to positive integer powers, expand the expression and substitute $i^2 = -1$.
Consider the example $(2+i)^{-4}$. The most efficient approach is to first square $(2+i)$, square the result to obtain the fourth power, and finally calculate the reciprocal. To simplify the resulting fraction, multiply by the conjugate of the denominator.
First, square the binomial: $$ (2+i)^2 = 4 + 4i + i^2 = 3 + 4i. $$ Next, square the result to find the fourth power: $$ (2+i)^4 = (3+4i)^2 = 9 + 24i + 16i^2 = -7 + 24i. $$ Then, apply the negative exponent by taking the reciprocal: $$ (2+i)^{-4} = \dfrac{1}{(2+i)^4} = \dfrac{1}{-7+24i}. $$ Finally, rationalize the denominator (as detailed in the Complex Division section): \begin{align*} \dfrac{1}{-7+24i} \cdot \dfrac{-7-24i}{-7-24i} & = \dfrac{-7-24i}{(-7)^2 + (24)^2} \\ & = \dfrac{-7-24i}{625}. \end{align*}
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To find a polynomial equation given a complex root, recall that for polynomials with real coefficients, complex roots occur in conjugate pairs. Thus, if a complex number is a solution, its conjugate is also a solution. To construct the equation, subtract each root from $x$ to form binomial factors and calculate their product.\noindent For example, if $2-i$ is a solution, then $2+i$ is also a solution. Multiplying the corresponding factors yields: $$ (x-(2-i))(x-(2+i)) = ((x-2)+i)((x-2)-i). $$ Recognizing this as a difference of squares simplifies the expansion: $$ (x-2)^2 - i^2 = (x^2-4x+4) - (-1) = x^2-4x+5. $$ Note that the imaginary terms always cancel when multiplying factors derived from conjugate pairs. This process is analogous to finding equations with irrational roots.
Consider a problem asking for a fourth-degree polynomial with integer coefficients and roots $2+i$ and $3+i$. The conjugate roots must be $2-i$ and $3-i$. Multiply the quadratic factors corresponding to each pair: \begin{align*} P(x) = & \ [(x-2-i)(x-2+i)] \\ & \cdot [(x-3-i)(x-3+i)] \\ = & \ (x^2-4x+5)(x^2-6x+10) \\ = & \ x^4-6x^3+10x^2-4x^3+24x^2-40x \\ & + 5x^2-30x+50 \\ = & \ x^4-10x^3+39x^2-70x+50. \end{align*}
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For polynomial equations of degree three or higher, factor the expression to isolate quadratic factors. Then, apply the quadratic formula to determine the complex solutions. For quadratic equations, use the quadratic formula directly or the method of completing the square.
When using the quadratic formula, a negative discriminant (the value under the radical) yields an imaginary term and, consequently, a complex solution. Similarly, when completing the square, equating a perfect square to a negative constant requires taking the square root of a negative number.
For example, solve $x^2 + 8x + 20 = 0$. Using the quadratic formula: \begin{align*} x & = \dfrac{-8 \pm \sqrt{8^2 - 4(1)(20)}}{2} \\ & = \dfrac{-8 \pm \sqrt{64 - 80}}{2} \\ & = \dfrac{-8 \pm \sqrt{-16}}{2} \\ & = \dfrac{-8 \pm 4i}{2} \\ & = -4 \pm 2i. \end{align*} Alternatively, by completing the square: \begin{align*} & x^2 + 8x = -20 \\ \implies & x^2 + 8x + 16 = -20 + 16 \\ \implies & (x + 4)^2 = -4 \\ \implies & x + 4 = \pm 2i \\ \implies & x = -4 \pm 2i. \end{align*}
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Complex numbers can be plotted on the complex plane, where the real part corresponds to the $x$-coordinate and the imaginary part corresponds to the $y$-coordinate. Some honors precalculus classes cover ways to convert these coordinates to polar form and use trigonometry to find roots of complex numbers. However, for this test, you only need to calculate the distance between complex numbers.
The distance between two complex numbers $a+bi$ and $c+di$ is determined by the distance formula: $$ \text{Distance} = \sqrt{(a-c)^2+(b-d)^2}. $$
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Answer key: 1.A, 2.B, 3.C, 4.D, 5.B, 6.C, 7.B, 8.A, 9.C, 10.D, 11.C, 12.C, 13.C, 14.C,
15.D, 16.B, 17.D, 18.D, 19.D,
20.C, 21.D, 22.C, 23.A, 24.C,
25.B, 26.C, 27.B, 28.C, 29.D, 30.A, 31.C, 32.B
