Formulas You’ll Need for the PSAT/NMSQT® Test

What do you need to succeed on the math section of the PSAT/NMSQT® Test?

Everything can be summarized in two sentences:

1. Know your calculator.
2. Know your formulas.

Why?

Well, you are going to be able to use a calculator during the PSAT/NMSQT® Math Test-Calculator, but if you don’t know how to use it to get the results you want accurately and quickly, you probably won’t have the time to solve all the questions in that section.

Do I Need to Know Formulas?

For both the Calculator and the No Calculator sections, you’ll need to know the formulas. Most of them are going to be given to you, but you need to know when to use them, and how to manipulate them to get the results you need.

What’s the Best Way to Prepare?

We are here to help you! Below, you’ll find a formula chart with the essential formulas in math that you are probably going to use during your PSAT/NMSQT® Math Test.

Why don’t you start becoming familiar with them by solving the free sample exercises we have for you here?

Category	Formula	Symbols	Comment
Heart of Algebra	\(x+a = b \rightarrow x = b-a\) \(x-a = b \rightarrow x= b+a\) \(x \cdot a = b \rightarrow x = b \div a\) \(x\div a = b \rightarrow x = b \cdot a\) \(x^a = b \rightarrow x = \sqrt[a]{b}\) \(\sqrt[a]{x}=b \rightarrow x = b^a\) \(a^x = b \rightarrow \dfrac{\log{b}}{\log{a}}\)	\(a,b\) = constants \(x\) = variable
Heart of Algebra	\(Ax+By=C\)	\(A, B, C\)= Constants \(y\) = dependent variable \(x\) = independent variable	Standard Form
Heart of Algebra	\(y=mx+b\)	\(m\) = slope \(b\) = intercept \(y\) = dependent variable \(x\) = independent variable	Slope-Intercept Form
Heart of Algebra	\(y-y_1 = m(x-x_1)\)	\(m\) = slope of a line \(y_n\) = dependent variable (point n) \(x_n\) = independent variable (point n)	Point-Slope Form
Heart of Algebra	\(m = \dfrac{y_2-y_1}{x_2-x_1}\)	\(m\) = slope of a line \(y_n\) = dependent variable (point n) \(x_n\) = independent variable (point n)	Slope of a line
Heart of Algebra	\(f(x) = ax^2 + bx + c\)	\(a,b\) = constants \(c\) = constant (y-axis intercept) \(x\) = variable	Standard form of Quadratic
Heart of Algebra	\(f(x) = a(x-h)^2 + k\)	\(a\) = constant \(h\) = constant (horizontal shift) \(k\) = constant (vertical shift) \(x\) = variable	Vertex form of Quadratic
Heart of Algebra	\(x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\)	\(a,b\) = constants from Standard form \(c\) = constant (y-axis intercept) \(x\) = x-intercept	Quadratic formula
Heart of Algebra	\(x = \frac{-b}{2a}\)	\(a,b\) = constants from Standard form \(x\) = axis of symmetry	Axis of symmetry
Heart of Algebra	\(P=4s\)	\(P\) = Perimeter of a square \(s\) = side length
Heart of Algebra	\(P = 2l + 2w\)	\(P\) = Perimeter of a rectangle \(l\) = length \(w\) = width
Heart of Algebra	\(P = s_1 + s_2 + s_3\)	\(P\) = Perimeter of a triangle \(s_n\) = side length
Heart of Algebra	\(C = 2 \pi r = \pi d\)	\(C\) = Perimeter of a circle \(r\) = radius \(d\) = diameter \(\pi \approx 3.14\)
Heart of Algebra	\(A = s^2\)	\(A\) = Area of a square \(s\) = side length
Heart of Algebra	\(A = lw\)	\(A\) = Area of a rectangle \(l\) = length \(w\) = width
Heart of Algebra	\(A = bh\)	\(A\) = Area of a parallelogram \(b\) = base \(h\) = height
Heart of Algebra	\(A = \frac{1}{2}bh\)	\(A\) = Area of a triangle \(b\) = base \(h\) = height
Heart of Algebra	\(A = h \cdot \dfrac{b_1+b_2}{2}\)	\(A\) = Area of a trapezoid \(b_n\) = base n \(h\) = height
Heart of Algebra	\(A = \pi r^2\)	\(A\) = Area of a circle \(r\) = radius \(\pi \approx 3.14\)
Heart of Algebra	\(V=lwh\)	\(V\) = Volume of a rectangular prism \(l\) = length \(w\) = width \(h\) = height
Heart of Algebra	\(V=Bh\)	\(V\) = Volume of a right prism \(B\) = area of the base \(h\) = height
Heart of Algebra	\(V = \pi r^2 h\)	\(V\) = Volume of a cylinder \(r\) = radius \(h\) = height \(\pi \approx 3.14\)
Heart of Algebra	\(V = \frac{1}{3}Bh\)	\(V\) = Volume of a pyramid \(B\) = area of the base \(h\) = height
Heart of Algebra	\(V = \frac{1}{3}\pi r^2 h\)	\(V\) = Volume of a cone \(r\) = radius \(h\) = height \(\pi \approx 3.14\)
Heart of Algebra	\(V = \frac{4}{3} \pi r^3\)	\(V\) = Volume of a sphere \(r\) = radius \(\pi \approx 3.14\)
Advanced Math	\((a \pm b)^2 = a^2 \pm 2ab + b^2\)	\(a,b\) = constants or variables	Square of a sum or difference
Advanced Math	\(a^2-b^2 = (a+b)\cdot(a-b)\)	\(a,b\) = constants or variable	Difference of squares
Advanced Math	\(a^3 - b^3 = (a-b)\cdot(a^2 + ab + b^2)\)	\(a,b\) = constants or variables	Difference of cubes
Advanced Math	\(a^3+b^3 = (a+b) \cdot (a^2 - ab + b^2)\)	\(a,b\) = constants or variables	Difference of cubes
Advanced Math	\(\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}\)	\(a,b,c,d\) = any real number	Remember to simplify the fraction (if possible)
Advanced Math	\(\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\)	\(a,b,c,d\) = any real number	Remember to simplify the fraction (if possible)
Advanced Math	\(\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}\)	\(a,b,c,d\) = any real number	Remember to simplify the fraction (if possible)
Advanced Math	\(a \frac{b}{c} = \frac{ac+b}{c}\)	\(a,b,c\) = any real number	Remember to simplify the fraction (if possible)
Advanced Math	\(x^a \cdot x^b = x^{a+b}\)	\(a,b,x\) = any real number
Advanced Math	\(\dfrac{x^a}{x^b} = x^{a-b}\)	\(a,b,x\) = any real number
Advanced Math	\((x^a)^b = x^{a\cdot b}\)	\(a,b,x\) = any real number
Advanced Math	\((xy)^a = x^a \cdot y^a\)	\(a,x,y\) = any real number
Advanced Math	\(x^1 = x\)	\(x\) = any real number
Advanced Math	\(x^0 = 1\)	\(x\) = any real number (\(x\ne 0\))
Advanced Math	\(x^{-a} = \frac{1}{x^a}\)	\(a,x\) = any real number (\(x\ne 0\))
Advanced Math	\(x^{\frac{a}{b}} = \sqrt[b]{x^a} = (\sqrt[b]{x})^a\)	\(a,b,x\) = any real number
Advanced Math	\(a^x = b \rightarrow \log _a b = x\)	\(a,b,x\) = any real number
Advanced Math	\(\ln (x) = \log _e x\)	\(x\) = any real number \(e \approx 2.718\) = Euler’s number
Advanced Math	\(a^{\log_a x} = x\)	\(a,x\) = any real number
Advanced Math	\(\log (a\cdot b) = \log (a) + \log (b)\)	\(a,b\) = any real number
Advanced Math	\(\log (a\div b) = \log (a) - \log (b)\)	\(a,b\) = any real number
Advanced Math	\(\log (a^b) = b \cdot \log (a)\)	\(a,b\) = any real number
Advanced Math	\(\log_a x = \log_b x \cdot \log_a b\)	\(a,b,x\) = any real number
Advanced Math	\(\log_a b = \dfrac{\log_x b}{\log_x a}\)	\(a,b,x\) = any real number
Advanced Math	\(\log_a a = 1\)	\(a\) = any real number (\(a\ne 0\))
Advanced Math	\(\log (1) = 0\)
Problem Solving and Data Analytics	\(a\cdot b \% = a \cdot \frac{b}{100}\)	\(a\) = any real number \(b \%\) = any percent	Remember to simplify (if possible)
Problem Solving and Data Analytics	\(\% = \dfrac{\lvert b - a \rvert}{b} \cdot 100 = \frac{c}{b} \cdot 100\)	\(\% = \%\) increase or decrease \(a\) = new value \(b\) = original value \(c\) = amount of change
Problem Solving and Data Analytics	\(\overline{x} = \dfrac{\Sigma x_i}{n}\)	\(\overline{x}\) = mean \(x_i\) = value of each measurement \(n\) = number of measurements
Problem Solving and Data Analytics	\(Md = (\frac{n+1}{2})^{th}\) term	\(Md\) = Median \(n\) = number of measurements (odd)
Problem Solving and Data Analytics	\(Md = \dfrac{(\frac{n}{2})^{th} \text{ term } + (\frac{n}{2} + 1)^{th} \text{ term}}{2}\)	\(Md\) = Median \(n\) = number of measurements (even)
Problem Solving and Data Analytics	\(s = \sqrt{\dfrac{\Sigma (x_i - \overline{x})^2}{n-1}}\)	\(s\) = standard deviation \(\overline{x}\) = mean \(x_i\) = value of each measurement \(n\) = number of measurements
Problem Solving and Data Analytics	\(V = s^2\)	\(V\) = Variance \(s\) = standard deviation
Problem Solving and Data Analytics	\(CV = RSD = 100 \cdot s \div \overline{x}\)	\(CV\) = Coefficient of variation \(RSD\) = Relative standard deviation \(s\) = standard deviation
Problem Solving and Data Analytics	\(p=\frac{d}{t}\)	\(p\) = probability of an event \(d\) = desired event \(t\) = total number of possible events