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Formulas: algebra, grade 9

The laws of powers

  • Multiplying powers with the same base
    am⋅an\displaystyle a^{m}\cdot a^{n} =\displaystyle = am+n\displaystyle a^{m+n}
  • Dividing powers with the same base
    am:an\displaystyle a^{m}:a^{n} =\displaystyle = am−n\displaystyle a^{m-n}a≠0\displaystyle a\neq0
  • A power of a power
    (am)n\displaystyle \left(a^{m}\right)^{n} =\displaystyle = am⋅n\displaystyle a^{m\cdot n}
  • A power of a product
    (a⋅b)n\displaystyle (a\cdot b)^{n} =\displaystyle = an⋅bn\displaystyle a^{n}\cdot b^{n}
  • A power of a quotient
    (ab)n\displaystyle \left(\frac{a}{b}\right)^{n} =\displaystyle = anbn\displaystyle \frac{a^{n}}{b^{n}}b≠0\displaystyle b\neq0
  • The power zero
    a0\displaystyle a^{0} =\displaystyle = 1\displaystyle 1a≠0\displaystyle a\neq0

The laws of square roots

  • The root of a product
    a⋅b\displaystyle \sqrt{a\cdot b} =\displaystyle = a⋅b\displaystyle \sqrt{a}\cdot\sqrt{b}a≥0, b≥0\displaystyle a\ge0,\ b\ge0
  • The root of a quotient
    ab\displaystyle \sqrt{\frac{a}{b}} =\displaystyle = ab\displaystyle \frac{\sqrt{a}}{\sqrt{b}}a≥0, b>0\displaystyle a\ge0,\ b>0
  • The square of a root
    (a)2\displaystyle \left(\sqrt{a}\right)^{2} =\displaystyle = a\displaystyle aa≥0\displaystyle a\ge0
  • The root of a square
    a2\displaystyle \sqrt{a^{2}} =\displaystyle = a\displaystyle aa≥0\displaystyle a\ge0

The extended distributive law and the short multiplication formulas

  • The extended distributive law
    (a+b)(c+d)\displaystyle (a+b)(c+d) =\displaystyle = ac+ad+bc+bd\displaystyle ac+ad+bc+bd
  • The square of a sum
    (a+b)2\displaystyle (a+b)^{2} =\displaystyle = a2+2ab+b2\displaystyle a^{2}+2ab+b^{2}
  • The square of a difference
    (a−b)2\displaystyle (a-b)^{2} =\displaystyle = a2−2ab+b2\displaystyle a^{2}-2ab+b^{2}
  • The difference of squares
    (a+b)(a−b)\displaystyle (a+b)(a-b) =\displaystyle = a2−b2\displaystyle a^{2}-b^{2}

The parabola

  • Standard form
    y=ax2+bx+c\displaystyle y=ax^{2}+bx+c
  • Vertex form – the vertex is (b,k)\displaystyle (b,k)
    y=a(x−b)2+k\displaystyle y=a(x-b)^{2}+k
  • Product form – it meets the x\displaystyle x-axis at x1\displaystyle x_1 and x2\displaystyle x_2
    y=a(x−x1)(x−x2)\displaystyle y=a(x-x_1)(x-x_2)
  • The vertex’s x\displaystyle x, in standard form
    x=−b2a\displaystyle x=-\frac{b}{2a}

The quadratic equation

  • The roots of ax2+bx+c=0\displaystyle {ax^{2}+bx+c}={0}, when a≠0\displaystyle a\neq0
    x1,2=−b±b2−4ac2a\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}