Algebra
Algebra on NYS mathematics exams tests structure: rewriting expressions, factoring, solving equations and inequalities, systems, and simple exponential models. Questions often require choosing an equivalent form or justifying each solution.
Core rules and concepts
- Factor by GCF first, then difference of squares, then trinomial patterns. The zero-product property says if ab=0, then a=0 or b=0.
- For ax^2+bx+c=0, the quadratic formula is x=(-b ± sqrt(b^2-4ac))/(2a); the discriminant b^2-4ac tells the number of real solutions.
- Solve systems by substitution or elimination. When multiplying or dividing an inequality by a negative, reverse the inequality sign.
- Keep domain restrictions for rational expressions; clear denominators only after noting excluded values. Check radical solutions for extraneous roots.
- Exponential models y=ab^x have constant multiplicative change; b>1 growth, 0<b<1 decay.
Worked example
Solve 2x^2-5x-3=0. Find ac=2(-3)=-6 and two numbers with product -6 and sum -5: -6 and 1. Rewrite the middle term: 2x^2-6x+x-3=0. Factor by grouping: 2x(x-3)+1(x-3)=0, so (2x+1)(x-3)=0. Thus x=-1/2 or x=3. Substitution confirms both satisfy the original equation.
Traps that cost marks
- Dividing both sides by a variable and losing the solution x=0.
- Sign errors when factoring or applying the quadratic formula.
- Forgetting that a rational equation’s solution may be excluded by the denominator.
- Reversing an inequality incorrectly, or forgetting to check a radical solution in the original equation.