Algebra
The Algebra domain is the largest strand on MCAS Grade 10. It tests polynomial and exponential expressions, linear and quadratic equations, inequalities, and systems, with emphasis on rewriting expressions and justifying each solution step.
Core rules
- Polynomial arithmetic: distribute carefully, combine like terms, and factor out the greatest common factor first.
- Factoring patterns: difference of squares a² − b² = (a − b)(a + b); perfect-square trinomials; trinomials by finding two numbers with the right product and sum.
- Quadratic equations: try factoring, then completing the square or the quadratic formula x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac tells the number of real solutions.
- Systems: substitution or elimination; parallel lines give no solution, identical lines give infinitely many.
- Inequalities: multiplying or dividing by a negative number reverses the inequality sign.
- Exponential expressions: recognize growth/decay factors and rewrite terms with a common base.
Worked example
Solve 2x² − 7x + 3 = 0. The product ac = 6 and the sum is −7, so the split terms are −6 and −1. Rewrite: 2x² − 6x − x + 3 = 0. Group: 2x(x − 3) − 1(x − 3) = 0, so (2x − 1)(x − 3) = 0. Therefore x = 1/2 or x = 3. Substituting either value into the original equation verifies the result.
Traps
- Dividing both sides by a variable can delete the solution x = 0.
- Squaring both sides can introduce extraneous solutions; check in the original.
- Watch signs when distributing a negative and when factoring.
- Reverse the inequality only for multiplication or division by a negative, not for adding or subtracting.