Wolfram Language Fast Introduction for Math Students
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Algebra

You can factor or expand algebraic expressions:

(Use CTRL+6 for typeset exponents.)
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Factor[x^2 + 2 x + 1]
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The Wolfram Language uses == (two equal signs) to test equality:

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2 + 2 == 4
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Combine algebraic expressions with == to represent an equation:

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1 + z == 15
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Commands like Solve find exact solutions to equations:

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Solve[x^2 + 5 x - 6 == 0, x]
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For approximate results, use NSolve:

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NSolve[7 x^2 + 3 x - 5 == 0, x]
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Pass in a system of equations as a list:

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Solve[{x^2 + 5 == y, 7 x - 5 == y}, {x, y}]
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Find the roots of an equation:

(|| is the symbol for Or.)
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Roots[x^2 + 3 x - 4 == 0, x]
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If a polynomial is not easily factorable, approximate results may be more useful:

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NRoots[360 + 234 x - 1051 x^2 + 11 x^3 + 304 x^4 - 20 x^5 == 0, x]
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The Reduce command reduces a set of inequalities into a simple form:

(Type <= for the ≤ symbol.)
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Reduce[{0 < x < 2, 1 <= x <= 4}, x]
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The reduced form may include multiple intervals:

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Reduce[(x - 1) (x - 2) (x - 3) (x - 4) > 0, x]
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NumberLinePlot is a handy way to visualize these results:

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NumberLinePlot[x < 1 || 2 < x < 3 || x > 4, {x, -10, 10}]
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Many equations and formulas are available through natural-language input:

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quadratic equation
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QUICK REFERENCE: Polynomial Equations »

QUICK REFERENCE: Equation Solving »