Algebra
Algebra is the largest domain on the Regents Algebra I exam. It covers linear equations and inequalities, systems, polynomial operations, factoring, quadratic equations, and exponential expressions. Many items are multi-step: write an equation from a context, solve it, and interpret the solution. The exam rewards showing equivalent forms and checking whether a solution satisfies the original condition.
Core rules
- Solve equations by inverse operations. When you multiply or divide an inequality by a negative number, reverse the inequality sign.
- A system's solution is the ordered pair that satisfies both equations. Substitution works well when one variable is isolated; elimination works when coefficients are opposites or easy multiples.
- For polynomials, distribute every term and combine like terms. (a+b)^2 = a^2+2ab+b^2, not a^2+b^2.
- For a quadratic, set the equation equal to zero before factoring. The zero-product property gives each solution. The quadratic formula solves ax^2+bx+c=0 when factoring is impractical; the discriminant b^2−4ac tells how many real solutions exist.
Worked example
Solve 3x^2−5x−2=0. The product ac is −6; find factors of −6 that add to −5: −6 and 1. Rewrite: 3x^2−6x+x−2=0. Group: 3x(x−2)+1(x−2)=0, so (3x+1)(x−2)=0. Thus x=−1/3 or x=2. Check x=2: 3(4)−10−2=0.
Traps
Do not divide both sides by a variable that could be zero; you will lose a solution. Watch signs when distributing a negative. In systems, a common error is solving for one variable and forgetting the other. Always substitute back into the original equation, not a rearranged one that may have introduced an error.