ACT Math Quadratics, Exponents and Matrices

ACT Math groups quadratics, polynomials, exponents, logarithms, matrices, and complex numbers under three content categories: Algebra, Functions, and Number and Quantity. These topics reward pattern recognition over memorization. A factored quadratic reveals its roots instantly. An exponent rule turns a messy expression into one term. A matrix packs a grid of numbers into one object you handle in a single step.

This guide breaks down each skill area with the rules tested, common problem setups, and original practice examples. Work through the sections in order to build from polynomials up to matrices and complex numbers, or jump to the topic slowing you down. Three free drills wait at the end: Quadratics and Polynomials, Logarithms and Exponentials, and Matrices and Complex Numbers.

Where These Topics Fit on the ACT Math Test

ACT Math has 45 questions in 50 minutes, including 4 embedded field-test items, which are unscored, leaving 41 scored items. A calculator is allowed on the mathematics section. Quadratics, polynomials, exponents, logarithms, matrices, and complex numbers spread across three reporting categories rather than one.

The table below shows each category's weight and what it covers for these topics.

Together these three categories account for 18 to 21 of the 41 scored items, 44 to 52 percent of the scored questions. All three sit inside the wider Preparing for Higher Math group, which covers 33 items, 80 percent of the scored questions. The rest of the group falls outside this guide.

Factoring and the Quadratic Formula

A quadratic equation has the form ax squared plus bx plus c equals zero. Two methods solve every quadratic on the test: factoring when the numbers cooperate, and the quadratic formula when they do not.

Factoring by Grouping and Common Patterns

Factor a quadratic by finding two numbers multiplying to a times c and adding to b, then split the middle term and group. Recognize three shortcut patterns before grouping.

• Difference of squares: a squared minus b squared factors to the quantity a plus b times the quantity a minus b.
• Perfect square trinomial: a squared plus 2ab plus b squared factors to the quantity a plus b, squared.
• Simple trinomial: x squared plus bx plus c factors to two binomials when two numbers multiply to c and add to b.

Example: factor x squared plus 5x plus 6. Two numbers multiplying to 6 and adding to 5 are 2 and 3, so the expression factors to the quantity x plus 2 times the quantity x plus 3. Setting each factor to zero gives roots of negative 2 and negative 3.

The Quadratic Formula

When a quadratic will not factor with integers, use the quadratic formula: x equals the quantity negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.

Example: solve 2x squared plus 3x minus 5 equals 0. Here a equals 2, b equals 3, and c equals negative 5. The discriminant, b squared minus 4ac, equals 9 minus negative 40, which is 49. The square root of 49 is 7, so x equals the quantity negative 3 plus or minus 7, divided by 4. The roots are x equals 1 and x equals negative 2.5.

Reading the Discriminant

The discriminant, the expression under the square root, tells you how many real roots a quadratic has before you finish solving. A positive discriminant gives two real roots. A discriminant of zero gives one repeated real root. A negative discriminant gives no real roots, only complex ones.

Polynomial Operations

Polynomials combine through addition, subtraction, multiplication, and division. ACT questions test each operation directly and inside word problems about area or combined expressions.

Adding, Subtracting, and Multiplying Polynomials

Add or subtract polynomials by combining like terms, matching each term's variable and exponent. Multiply polynomials by distributing every term in the first factor across every term in the second. Example: multiply the quantity x plus 4 times the quantity x minus 3. Distributing gives x squared minus 3x plus 4x minus 12, which simplifies to x squared plus x minus 12.

Dividing Polynomials

Divide a polynomial by a monomial by dividing each term separately. Divide a polynomial by a binomial using long division or synthetic division, matching the process used for dividing whole numbers. Example: divide 6x squared plus 9x by 3x. Dividing each term gives 2x plus 3.

Exponent and Logarithm Rules

Exponent rules simplify expressions with powers, and logarithms reverse exponential relationships. The two skills work as a pair: a logarithm answers the question an exponential equation asks.

Core Exponent Rules

Six rules cover nearly every exponent question on the test.

• Product rule: multiplying like bases adds the exponents, so x to the a times x to the b equals x to the a plus b.
• Quotient rule: dividing like bases subtracts the exponents, so x to the a divided by x to the b equals x to the a minus b.
• Power rule: raising a power to a power multiplies the exponents, so the quantity x to the a, to the b, equals x to the a times b.
• Zero exponent: any nonzero base raised to the zero power equals 1.
• Negative exponent: a negative exponent flips the base into a fraction, so x to the negative a equals 1 divided by x to the a.
• Rational exponent: a fractional exponent represents a root, so x to the 1 over n equals the nth root of x.

Example: simplify the quantity x to the third, times x to the negative 5. Adding the exponents gives x to the negative 2, which rewrites as 1 divided by x squared.

Logarithm Rules

A logarithm answers what exponent a base needs to reach a given value. Log base b of x equals y means b raised to the y power equals x. The same product, quotient, and power rules governing exponents govern logarithms in reverse.

Example: solve log base 2 of x equals 5. Rewriting in exponential form gives 2 to the 5th equals x, so x equals 32. Logarithmic functions are named directly in the Functions category, so expect at least one question asking you to evaluate, solve, or graph one.

Matrix Operations and Complex Number Arithmetic

Matrices organize numbers into rows and columns, and the Number and Quantity category names vectors and matrices, real and complex number systems, and integer and rational exponents. Complex numbers extend the real number line to include the square root of negative 1.

Adding, Subtracting, and Multiplying Matrices

Add or subtract matrices by combining entries in matching positions, which only works when both matrices share the same dimensions. Multiply two matrices by taking the dot product of each row in the first matrix with each column in the second, which requires the number of columns in the first matrix to match the number of rows in the second. Example: add the matrix with rows 1, 2 and 3, 4 to the matrix with rows 5, 6 and 7, 8. Adding matching entries gives a result matrix with rows 6, 8 and 10, 12.

Complex Number Arithmetic

The imaginary unit i equals the square root of negative 1, and i squared equals negative 1. Add or subtract complex numbers by combining real parts together and imaginary parts together. Multiply complex numbers by distributing each term, then replacing i squared with negative 1.

• Addition: the quantity a plus bi, plus the quantity c plus di, equals the quantity a plus c, plus the quantity b plus d, times i.
• Multiplication: the quantity a plus bi, times the quantity c plus di, equals ac plus adi plus bci plus bdi squared, which simplifies using i squared equals negative 1.

Example: multiply the quantity 2 plus 3i times the quantity 1 minus 4i. Distributing gives 2 minus 8i plus 3i minus 12i squared. Replacing i squared with negative 1 turns negative 12i squared into positive 12, so the result is 14 minus 5i.

Practice Quadratics, Exponents and Matrices Questions on 10exam

Three free drills on 10exam target the skills in this guide. The Quadratics & Polynomials drill covers factoring, the quadratic formula, and polynomial operations. The Logarithms & Exponentials drill covers exponent rules and logarithm rules. The Matrices & Complex Numbers drill covers matrix operations and complex number arithmetic. All three drills are free and untimed, with instant explanations, and no signup is required.

For full length, timed Math practice under real test conditions, take one of our free practice tests on the ACT exam page. Review every missed question from these topics, then return to the matching drill above. See our ACT Math section guide for the full format, calculator rules, and content breakdown across every category. If function notation or coordinate geometry trips you up, work through our functions and coordinate geometry guide next.

Sources

• ACT Reporting Category Tables Comparison (Legacy vs Enhanced) | ACT
• Description of Math Test | ACT