| Term | Definition | Rules / Outcomes |
|---|---|---|
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Rational Exponent
\(b^{m/n}\)
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A fractional exponent. The denominator is the index of the root; the numerator is the power applied to the base.
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\(b^{m/n} = \left(\sqrt[n]{(b)}\right)^m = \sqrt[n]{(b^m)}\)
Order: root and power can be applied in either order.
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Discriminant
\(b^2 - 4ac\)
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The expression under the radical in the quadratic formula. Determines the number and type of solutions to \(ax^2+bx+c=0\).
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If \(b^2-4ac > 0\): 2 real solutions
If \(b^2-4ac = 0\): 1 real solution
If \(b^2-4ac < 0\): 0 real solutions (2 complex)
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Imaginary Unit
\(i\)
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Defined so that \(i^2 = -1\). It is the square root of \(-1\). It is not a real number.
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\(\sqrt{(-a)} = \sqrt{(a)} \cdot i\) for \(a > 0\)
Replace every \(i^2\) in a product with \(-1\).
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Complex Number
\(a + bi\)
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Standard form with real part \(a\) and imaginary part \(bi\), where \(a\) and \(b\) are real numbers.
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Add/Subtract: combine real parts, then imaginary parts
Multiply: FOIL, then replace \(i^2\) with \(-1\)
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Square Root Equation
\(\sqrt{(f(x))} = k\)
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An equation with a variable inside a square root. Isolate the radical, then square both sides.
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If \(k \geq 0\): possible solution — check for extraneous
If \(k < 0\): no real solution (\(\sqrt{\phantom{x}}\) is never negative)
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Cube Root Equation
\(\sqrt[3]{(f(x))} = k\)
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An equation with a variable inside a cube root. Isolate the radical, then cube both sides.
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Always exactly 1 real solution for any value of \(k\)
No extraneous solutions.
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Completing the Square
\((x+p)^2 = q\)
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Rewriting a quadratic so one side is a perfect square trinomial. Then take square roots of both sides.
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If \(q > 0\): 2 real solutions \(x = -p \pm \sqrt{(q)}\)
If \(q = 0\): 1 real solution \(x = -p\)
If \(q < 0\): 2 complex solutions \(x = -p \pm i\sqrt{(-q)}\)
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Quadratic Formula
\(\dfrac{-b \pm \sqrt{(b^2-4ac)}}{2a}\)
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Gives all solutions to \(ax^2+bx+c=0\) directly from its coefficients. Works for every quadratic.
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Sign of \(b^2-4ac\) determines solution type (see Discriminant row)
When \(b^2-4ac < 0\): \(\sqrt{(b^2-4ac)} = i\sqrt{(|b^2-4ac|)}\)
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