Advanced Linear Systems and Quadratics
The SAT Math section (Calculator and No-Calculator combined) tests algebra at two levels. The 600–700 range tests standard linear equations and basic quadratics. The 700–800 range tests multi-variable systems, quadratic-linear systems, and applications requiring symbolic manipulation. Multi-variable systems: given a system of two equations with two unknowns (ax + by = c and dx + ey = f), the substitution and elimination methods produce unique solutions. Key SAT extensions: (1) No solution — a system with no solution has parallel lines: the ratio of x-coefficients equals the ratio of y-coefficients but not the ratio of constants (a/d = b/e ≠ c/f). (2) Infinitely many solutions — dependent system: all three ratios are equal (a/d = b/e = c/f). (3) Quadratic-linear system: substitute the linear equation's expression for y into the quadratic to produce a standard quadratic, solve for x, then substitute back. The discriminant (b²−4ac): if positive, two solutions; if zero, one solution (the line is tangent to the parabola); if negative, no real solutions. Advanced factoring: difference of squares (a²−b² = (a+b)(a−b)), perfect square trinomials (a²+2ab+b² = (a+b)²), and factoring by grouping. SAT higher-difficulty problems often test completing the square to find vertex form (f(x) = a(x−h)²+k where (h,k) is the vertex), which is required for problems asking for the minimum/maximum of a quadratic or the axis of symmetry.