Number of solutions to linear equations
- \(x=5\): one solution
- \(3=5\): no solutions
- \(5=5\): infinite solutions
Basic lines
Slope: rise over run, or \(\frac{\Delta y}{\Delta x}\)
Forms
- Slope–intercept: \(y=mx+b\) — e.g. \(y=5x+3\)
- Point–slope: \(y-b=m\left(x-a\right)\) — e.g. \(y-5=\frac{1}{2}\left(x-3\right)\)
- Standard: \(ax+by=c\) — e.g. \(5x+3y=15\)
Sequences: basic
- Arithmetic sequence: \(f\left(n\right)=10+5n\)
- Geometric sequence: \(f\left(n\right)=10\cdot5^n\)
Systems of equations
Methods for solving
- Elimination: add sides of the two equations together, when one can cancel the other out
Number of solutions
- \(0=20\): no solutions
- \(0=0\): infinite solutions
- Otherwise, if you find, say, \(y=5\) or some such, one solution
Exponents and square roots
Properties
- \(x^ax^b=x^{a+b}\)
- \(\dfrac{x^a}{x^b}=x^{a-b}\)
- \(\left(x^a\right)^b=x^{ab}\)
- \(\left(xy\right)^a=x^ay^a\)
- \(x^ay^a=\left(xy\right)^a\)
- \(\left(x^b\cdot y^c\right)^a=\left(x^{b^a}\right)\left(y^{c^a}\right)=x^{ba}y^{ca}\)
- \(\left(\dfrac{x}{y}\right)^a=\left(\dfrac{x^a}{y^a}\right)\)
Other notes
- \(p^2=0.81 \rightarrow p=\pm \sqrt{0.81}\)
- \((\sqrt[b]{a})^c = \sqrt[b]{a^c}\)
Square roots
- \(\sqrt{xy}=\sqrt{x}\sqrt{y}\)
- \(\sqrt{x+y}\ne\sqrt{x}+\sqrt{y}\)
- \(\sqrt{\dfrac{x}{y}}=\dfrac{\sqrt{x}}{\sqrt{y}}\)
- \(a\sqrt{x}\pm b\sqrt{x} = (a\pm b)\sqrt{x}\)
- \(a\sqrt{x} = \sqrt{a^2 \cdot x}\)
Factorization
- Distributive property: factoring out a common factor
- Splitting the middle term
- Factoring by grouping
Splitting the middle term
- Given some quadratic like \(x^2-3x-10\), split it into \(x^2+\left(a+b\right)x+ab\) to obtain the form\(\left(x+a\right)\left(x+b\right)\) such that \(a+b\) is the middle term coefficient and \(ab\) is the last term
Factoring by grouping
- Example: \(4x^2+25x-21\). Find \(a\cdot b=4\cdot-21=-84\) and \(a+b=25\), the middle element.
- Then change the factoring to \(\left(4x^2+28x\right)-\left(3x-21\right)\) and factor out \(\left(x+7\right)\).
Perfect square form
- \(\left(Ax+B\right)^2=A^2x^2+2ABx+B^2\)
Solving quadratics
- Take the square root
- If equals 0, then find the roots
- Turn it into a "completing the square" problem by dividing by leading coefficient; then take half of the middle coefficient, square it, and make that the third
- Trick for solving things that don't seem to have a good solution: variable replacement. e.g. from \(\left(2x-3\right)^2=2\left(2x-3\right)\), substitute \(p=2x-3\) and find \(p^2=2p\).
- Vertex of a parabola at \(\frac{-b}{2a}\)
- When solving square-root/exponent quadratics, when raising both sides of an equation to an even power, check for extraneous solutions
Forms
- \(3\left(x+1\right)\left(x+11\right)\): factored form (for finding zeroes)
- \(3\left(x-6\right)^2+75\): vertex form (for finding the vertex)
- \(3x^2+36x+33\): standard form (for finding y-intercept)
Quadratic formula
- \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Discriminant
\(b^2-4ac\)
- \(>0\): 2 solutions
- \(=0\): 1 solution
- \(<0\): no solutions