Teacher License – Math 1 Complete Review Guide 2026–2027

4. Key identities: \(a^{2}-b^{2}=(a-b)(a+b)\) and \(a^{2}\pm 2ab+b^{2}=(a\pm b)^{2}\).
5. To factor \(x^{2}+bx+c\) look for two numbers with product \(c\) and sum \(b\), so \(x^{2}+5x+6=(x+2)(x+3)\).
6. To simplify an algebraic fraction, factor the numerator and denominator, cancel the common factor, and exclude values that make the denominator zero.
7. Solve a linear equation by undoing operations in reverse order: \(3x+4=19\) gives \(3x=15\), so \(x=5\).
8. The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\); the sum of the roots is \(-\frac{b}{a}\) and their product is \(\frac{c}{a}\).
9. The discriminant \(b^{2}-4ac\): positive gives two different real solutions, zero gives one repeated real solution, and negative gives no real solution.
10. For \(a>0\), \(|x|<a\) means \(-a<x<a\), and \(|x|>a\) means \(x<-a\) or \(x>a\).
11. A system of two linear equations: eliminate one variable by adding or subtracting, or substitute, then put the value back to find the other; the solution is the point where the lines meet.
12. Parallel lines have the same slope and different intercepts, so their system has no solution; coinciding lines share slope and intercept, so the system has infinitely many solutions.
13. Arithmetic sequence: \(a_{n}=a_{1}+(n-1)d\), with sum of \(n\) terms \(S_{n}=\frac{n(a_{1}+a_{n})}{2}\); geometric sequence: \(a_{n}=a_{1}r^{n-1}\).
14. If the second differences of a pattern are constant, the pattern is quadratic, like \(1, 4, 9, 16\) with general term \(n^{2}\).
15. A relation is a function if each element of the domain is paired with exactly one element of the range; graphically this is the vertical line test.
16. The linear function \(f(x)=mx+b\) has slope \(m\) and y-intercept \(b\); the quadratic \(f(x)=ax^{2}+bx+c\) has a parabola that opens upward when \(a>0\).
17. Adding or subtracting two linear functions gives a linear function, and multiplying them gives a quadratic; for \(\frac{f(x)}{g(x)}\) we need \(g(x)\ne 0\).

Common mistakes in this chapter

• ✕ The complement \(A'\) is always taken inside the universal set \(U\), while \(A-B\) means the elements in \(A\) but not in \(B\); in general \(A-B\ne B-A\).
• ✕ Forgetting to subtract \(n(A\cap B)\) when counting a union: adding only \(n(A)+n(B)\) counts the shared elements twice.
• ✕ The common slip \((a+b)^{2}=a^{2}+b^{2}\); the correct form is \((a+b)^{2}=a^{2}+2ab+b^{2}\), and likewise \((a-b)^{2}=a^{2}-2ab+b^{2}\).
• ✕ Multiplying or dividing by a negative number reverses the inequality: \(-2x>6\) gives \(x<-3\), not \(x>-3\).
• ✕ In the quadratic formula people drop the sign of \(-b\) or divide by \(a\) instead of \(2a\); and \(x^{2}=9\) has two solutions, \(x=\pm 3\), not one.
• ✕ Not checking the answer in a system of equations, and mixing up "no solution" (parallel lines) with "infinitely many solutions" (the same line).
• ✕ For a sequence starting at \(a_{1}\) the general term is \(a_{1}+(n-1)d\), not \(a_{1}+nd\); the usual error is an off-by-one count.
• ✕ A function gives each input exactly one output: two inputs may share an output (like \(x^{2}\)), but one input may not have two outputs; the domain of \(\frac{1}{x-2}\) is all real numbers except \(2\).