Algebra 2 Honors Review Notes

Simplifying Roots and Expressions

Basic Square Roots

  • Square Roots:

    • 16=4\sqrt{16} = 4

    • 0.0001=0.01\sqrt{0.0001} = 0.01

  • Complex Roots:

    • −64x12=−4x4\sqrt{-64x^{12}} = -4x^{4}

    • Not a real number: −81\sqrt{-81}

  • Absolute Values:

    • When working with variables, always assume they can be any real number.

Rational and Irrational Forms

  • Simplifying Square Terms:

    • (x−5)2=∣x−5∣\sqrt{(x-5)^{2}} = |x-5|

    • x2+25\sqrt{x^{2} + 25} (already simplified)

  • Manipulating Powers:

    • x2−8x+16=(x−4)2=∣x−4∣\sqrt{x^{2}-8x+16} = \sqrt{(x-4)^{2}} = |x-4|

Function Evaluations

  • Functions:

    • Let f(x)=3x+2f(x) = \sqrt{3x + 2} and g(x)=x+2g(x) = \sqrt{x + 2}.

  • Evaluating Functions:

    • f(0)=3(0)+2=2f(0) = \sqrt{3(0) + 2} = \sqrt{2}

    • g(7)=7+2=9=3g(7) = \sqrt{7 + 2} = \sqrt{9} = 3

    • g(−10)=−10+2g(-10) = \sqrt{-10 + 2} = is not a real number.

Domain and Graphing

  • Identifying Domain:

    • For f(x)=x+2f(x) = \sqrt{x + 2}, the domain is [−2,∞)[-2, \infty).

    • For g(x)=5x+2g(x) = \sqrt{5x + 2}, the domain is (−∞,∞)(-\infty, \infty).

The Mosteller Formula

  • Body Surface Area:

    • B=h⋅w3131B = \sqrt{\frac{h \cdot w}{3131}}, where:

    • h=70h = 70 inches (height)

    • w=175w = 175 pounds (weight)

    • Calculation:

    • B=70⋅1753131B = \sqrt{\frac{70 \cdot 175}{3131}}

    • B≈1.98m2B \approx 1.98 m^{2}

Simplifying Additional Roots

  • Higher-Order Radicals:

    • 273=3\sqrt[3]{27} = 3 (from cube root simplification)

    • General simplification examples:

    • (81)3=27\left(\sqrt{81}\right)^{3} = 27

  • Roots of Negative Numbers:

    • −64=i64=8i\sqrt{-64} = i\sqrt{64} = 8i (imaginary number representation)

    • Not a real number: −100\sqrt{-100}

  • Use of Absolute Values:

    • For expressions involving non-real solutions, use absolute value principles.

Combining and Simplifying Radicals

  • Operations with Radicals:

    • Combine like radicals:

    • 48−275+312\sqrt{48} - 2\sqrt{75} + 3\sqrt{12}

      • Simplify to get final result.

Factoring and Expanding

  • Using Algebraic Identities:

    • (a+b)(a−b)=a2−b2(a + b)(a - b) = a^{2} - b^{2}

    • (x−2+3)(x−2−3)=9−4(\sqrt{x} - 2 + 3)(\sqrt{x} - 2 - 3) = 9 - 4

    • Keep expanding until reaching simplest form.

Final Notes

  • When simplifying, always ensure to account for domain restrictions of square roots.

  • Non-real results should be recorded appropriately, use imaginary units when necessary.

  • Ensure precision during evaluations, keeping radical forms where simplification may not yield integers.