5.2 Core Algebra and Geometry Workflows

5.2 Core Algebra and Geometry Workflows

MK rewards clean, repeatable procedures. Below are the workflows that cover the majority of items.

Solving linear equations

Isolate the variable by undoing operations in reverse PEMDAS order:

1. Clear parentheses (distribute).
2. Combine like terms on each side.
3. Move variable terms to one side, constants to the other.
4. Divide by the coefficient.

Worked example: Solve 3(x − 2) = 9. Distribute → 3x − 6 = 9. Add 6 → 3x = 15. Divide by 3 → x = 5.

Factoring and solving quadratics

Most MK quadratics factor cleanly — try factoring before the formula. For x² + bx + c, find two numbers that multiply to c and add to b.

Worked example: Solve x² + 5x + 6 = 0. Need factors of 6 summing to 5 → 2 and 3. So (x + 2)(x + 3) = 0, giving x = −2 or x = −3. If no integer factors exist, use the quadratic formula x = (−b ± √(b² − 4ac))/2a.

Exponents and radicals

Simplifying radicals: factor out perfect squares. √50 = √(25·2) = 5√2.

Multiplying binomials with FOIL

FOIL = First, Outer, Inner, Last. (x + 3)(x − 2) = x² (First) − 2x (Outer) + 3x (Inner) − 6 (Last) = x² + x − 6.

Core geometry

• Triangle area: ½ · base · height. A triangle with base 8, height 5 → area 20.
• Pythagorean theorem: a² + b² = c². Legs 3 and 4 → hypotenuse √25 = 5. Memorize 3-4-5 and 5-12-13 triples.
• Circle: area = πr², circumference = 2πr. Radius 4 → area 16π, circumference 8π.
• Rectangle/box: area = lw; volume of a rectangular solid = lwh.
• Angles: interior angles of a triangle sum to 180°; of a quadrilateral, 360°.

Back-solving: a legitimate shortcut

When four numeric answers are offered and solving is messy, plug choices into the original equation and pick the one that works. Start with the middle value to narrow direction. This is often faster than algebra on MK and is a sanctioned strategy, not cheating.

Inequalities and absolute value

Solve inequalities exactly like equations, with one rule: flip the inequality sign whenever you multiply or divide both sides by a negative number. Solving −2x < 6 → divide by −2 and reverse → x > −3. Absolute value means distance from zero, so |x| = 5 yields two answers, x = 5 and x = −5; |x − 3| = 4 gives x = 7 or x = −1. Forgetting the second solution is a frequent MK miss.

Systems of two equations

MK occasionally pairs two linear equations. Use substitution (solve one equation for a variable, plug into the other) or elimination (add or subtract the equations to cancel a variable). For x + y = 10 and x − y = 4, add them: 2x = 14 → x = 7, then y = 3. Elimination is usually the fastest when coefficients line up.

Simplifying algebraic fractions

Factor numerator and denominator, then cancel common factors. (x² − 9)/(x + 3) = (x + 3)(x − 3)/(x + 3) = x − 3. Recognizing the difference of squares pattern a² − b² = (a + b)(a − b) unlocks many of these items instantly. Two more patterns worth memorizing are perfect-square trinomials a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)².

Coordinate geometry quick facts

The distance between two points is √((x₂−x₁)² + (y₂−y₁)²); the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). Parallel lines share a slope; perpendicular lines have slopes that are negative reciprocals (a line with slope 2 is perpendicular to one with slope −½). These three relationships cover almost every MK graph item.

Working with function notation

MK sometimes uses f(x) notation, which simply means 'substitute the input for x.' If f(x) = 2x² − 3, then f(4) = 2(16) − 3 = 29. The notation looks intimidating but the operation is plain substitution followed by arithmetic in PEMDAS order. Composite notation f(g(x)) means evaluate g first, then feed its output into f — work from the inside out, exactly as with nested parentheses.

Proportional reasoning and scaling

Many geometry items hinge on scaling. If a shape's linear dimensions all double, its area quadruples (scales by 2²) and its volume scales by eight (2³). So doubling a cube's edge multiplies its volume by 8, not 2. This square-and-cube relationship between length, area, and volume is a quiet but recurring MK theme; a distractor will always offer the naive 'double' answer.

A disciplined solution routine

For any MK item, run the same four-step routine: (1) read the exact demand — solve for x, simplify, find the area, etc.; (2) choose the procedure that matches the structure (factor, distribute, apply a formula); (3) execute carefully, writing each step so a sign error is catchable; (4) sanity-check the result against an estimate and against what was asked. Skipping step one causes Trap 4 (answering the wrong quantity); skipping step four lets sign and decimal errors through.

Practicing the routine until it is automatic is what separates an 80% score from a 95% score, because on MK the math is rarely the obstacle — the discipline is.