ACT Mathematics — quadratics, systems, inequalities and the coordinate plane

This material covers the middle of the ACT mathematics section: quadratic equations, systems of two equations, inequalities, and everything that happens once points and lines are given coordinates. Arithmetic, percents and ratios are covered by the material on percents, ratios, rates and linear equations; plane geometry and trigonometry by the ones after it.

Twelve questions, two on each of six things. Two quadratics, one factored and one taken through the formula, and both asked in a form that has exactly one answer. That last point matters more than it sounds: an equation with two roots cannot be asked "what is the solution", because two of the four options would then be correct — so each asks for the larger root, and the smaller one sits in the list as a distractor for anyone who solves correctly and reports the wrong half. Two systems, one by elimination where the x terms cancel outright and one by substitution where an expression for y goes straight into the second equation. Two inequalities, the second turning on the one rule that has no counterpart in ordinary equations: dividing by a negative reverses the sign. Two on slope, two on the equation of a line and its intercepts, and two on distance and midpoint.

Every wrong option is a real mistake with a name. The slope option reading 1/2 is the fraction upside down, which describes a quite different line. The perpendicular option reading 1/3 takes the reciprocal and forgets the sign, and multiplying it by the original slope gives 1 rather than -1 — so the two lines it describes are not perpendicular at all. The line option reading y = 2x + 3 treats the y-value of a given point as the y-intercept, which is only ever true when that point sits on the y-axis. The midpoint option (2, 2) halves the differences instead of averaging the values, and so states how far the midpoint is from each end rather than where it is.

Each key is verified inside its own explanation by putting it back into the original equation, which is also the habit the material is trying to build: an answer that has been substituted back is an answer you no longer have to trust your arithmetic for. Twelve flashcards carry the quadratic formula, the choice between substitution and elimination, the reversal rule, slope from two points, the parallel and perpendicular conditions, point-slope and slope-intercept form, and the distance and midpoint formulas with the check that goes with each.

Zestly is an independent study tool. It is not affiliated with ACT, Inc., which owns the test, and it is not a test center.

• Solve a quadratic by factoring or by the formula, and check a root by substituting it back
• Choose elimination when a variable's coefficients already match, and substitution when one equation gives a variable on its own
• Report the variable that was actually asked for, rather than the one solved for first
• Reverse an inequality when and only when dividing or multiplying by a negative
• Find slope from two points with both differences taken in the same order, and get the sign right
• Recognize parallel and perpendicular slopes, and verify perpendicularity by checking that the product is -1
• Write the equation of a line from a point and a slope, and find both intercepts from the equation
• Apply the distance and midpoint formulas, and sanity-check a midpoint by seeing that it lies between the endpoints

Written for the algebra and coordinate-geometry strands of the ACT mathematics section, following the categories the test's own published description names: linear and quadratic equations, systems of equations, inequalities, and graphing points and lines in the coordinate plane. Every point, line and equation in the bank is stated in full inside its own question, so the material works as plain text; two coordinate questions also show their points on a grid. Nothing is taken from any ACT publication or released test.